text:
// Based on Eclipse Paho.
/*******************************************************************************
 * Copyright (c) 2014 IBM Corp.
 *
 * All rights reserved. This program and the accompanying materials
 * are made available under the terms of the Eclipse Public License v1.0
 * and Eclipse Distribution License v1.0 which accompany this distribution.
 *
 * The Eclipse Public License is available at
 *    http://www.eclipse.org/legal/epl-v10.html
 * and the Eclipse Distribution License is available at
 *   http://www.eclipse.org/org/documents/edl-v10.php.
 *
 * Contributors:
 *    Allan Stockdill-Mander/Ian Craggs - initial API and implementation and/or initial documentation
 *******************************************************************************/

/**
 * @file aws_iot_mqtt_client.c
 * @brief MQTT client API definitions
 */

#ifdef __cplusplus
extern "C" {
#endif

#include "aws_iot_log.h"
#include "aws_iot_mqtt_client_interface.h"

#ifdef _ENABLE_THREAD_SUPPORT_
#include "threads_interface.h"
#endif

const IoT_Client_Init_Params iotClientInitParamsDefault = IoT_Client_Init_Params_initializer;
const IoT_MQTT_Will_Options iotMqttWillOptionsDefault = IoT_MQTT_Will_Options_Initializer;
const IoT_Client_Connect_Params iotClientConnectParamsDefault = IoT_Client_Connect_Params_initializer;

ClientState aws_iot_mqtt_get_client_state(AWS_IoT_Client *pClient) {
	FUNC_ENTRY;
	if(NULL == pClient) {
		return CLIENT_STATE_INVALID;
	}

	FUNC_EXIT_RC(pClient->clientStatus.clientState);
}

#ifdef _ENABLE_THREAD_SUPPORT_
IoT_Error_t aws_iot_mqtt_client_lock_mutex(AWS_IoT_Client *pClient, IoT_Mutex_t *pMutex) {
	FUNC_ENTRY;
	IoT_Error_t threadRc = FAILURE;

	if(NULL == pClient || NULL == pMutex){
		FUNC_EXIT_RC(NULL_VALUE_ERROR);
	}

	if(false == pClient->clientData.isBlockOnThreadLockEnabled) {
		threadRc = aws_iot_thread_mutex_trylock(pMutex);
	} else {
		threadRc = aws_iot_thread_mutex_lock(pMutex);
		/* Should never return Error because the above request blocks until lock is obtained */
	}

	if(SUCCESS != threadRc) {
		FUNC_EXIT_RC(threadRc);
	}

	FUNC_EXIT_RC(SUCCESS);
}

IoT_Error_t aws_iot_mqtt_client_unlock_mutex(AWS_IoT_Client *pClient, IoT_Mutex_t *pMutex) {
	if(NULL == pClient || NULL == pMutex) {
		return NULL_VALUE_ERROR;
	}
	IOT_UNUSED(pClient);
	return aws_iot_thread_mutex_unlock(pMutex);
}
#endif

IoT_Error_t aws_iot_mqtt_set_client_state(AWS_IoT_Client *pClient, ClientState expectedCurrentState,
										  ClientState newState) {
	IoT_Error_t rc;
#ifdef _ENABLE_THREAD_SUPPORT_
	IoT_Error_t threadRc = FAILURE;
#endif

	FUNC_ENTRY;
	if(NULL == pClient) {
		FUNC_EXIT_RC(NULL_VALUE_ERROR);
	}

#ifdef _ENABLE_THREAD_SUPPORT_
	rc = aws_iot_mqtt_client_lock_mutex(pClient, &(pClient->clientData.state_change_mutex));
	if(SUCCESS != rc) {
		return rc;
	}
#endif
	if(expectedCurrentState == aws_iot_mqtt_get_client_state(pClient)) {
		pClient->clientStatus.clientState = newState;
		rc = SUCCESS;
	} else {
		rc = MQTT_UNEXPECTED_CLIENT_STATE_ERROR;
	}

#ifdef _ENABLE_THREAD_SUPPORT_
	threadRc = aws_iot_mqtt_client_unlock_mutex(pClient, &(pClient->clientData.state_change_mutex));
	if(SUCCESS == rc && SUCCESS != threadRc) {
		rc = threadRc;
	}
#endif

	FUNC_EXIT_RC(rc);
}

IoT_Error_t aws_iot_mqtt_set_connect_params(AWS_IoT_Client *pClient, IoT_Client_Connect_Params *pNewConnectParams) {
	FUNC_ENTRY;
	if(NULL == pClient || NULL == pNewConnectParams) {
		FUNC_EXIT_RC(NULL_VALUE_ERROR);
	}

	pClient->clientData.options.isWillMsgPresent = pNewConnectParams->isWillMsgPresent;
	pClient->clientData.options.MQTTVersion = pNewConnectParams->MQTTVersion;
	pClient->clientData.options.pClientID = pNewConnectParams->pClientID;
	pClient->clientData.options.clientIDLen = pNewConnectParams->clientIDLen;
	pClient->clientData.options.pUsername = pNewConnectParams->pUsername;
	pClient->clientData.options.usernameLen = pNewConnectParams->usernameLen;
	pClient->clientData.options.pPassword = pNewConnectParams->pUsername;
	pClient->clientData.options.passwordLen = pNewConnectParams->passwordLen;
	pClient->clientData.options.will.pTopicName = pNewConnectParams->will.pTopicName;
	pClient->clientData.options.will.topicNameLen = pNewConnectParams->will.topicNameLen;
	pClient->clientData.options.will.pMessage = pNewConnectParams->will.pMessage;
	pClient->clientData.options.will.msgLen = pNewConnectParams->will.msgLen;
	pClient->clientData.options.will.qos = pNewConnectParams->will.qos;
	pClient->clientData.options.will.isRetained = pNewConnectParams->will.isRetained;
	pClient->clientData.options.keepAliveIntervalInSec = pNewConnectParams->keepAliveIntervalInSec;
	pClient->clientData.options.isCleanSession = pNewConnectParams->isCleanSession;

	FUNC_EXIT_RC(SUCCESS);
}

IoT_Error_t aws_iot_mqtt_init(AWS_IoT_Client *pClient, IoT_Client_Init_Params *pInitParams) {
	uint32_t i;
	IoT_Error_t rc;
	IoT_Client_Connect_Params default_options = IoT_Client_Connect_Params_initializer;

	FUNC_ENTRY;

	if(NULL == pClient || NULL == pInitParams || NULL == pInitParams->pHostURL || 0 == pInitParams->port ||
	   NULL == pInitParams->pRootCALocation || NULL == pInitParams->pDevicePrivateKeyLocation ||
	   NULL == pInitParams->pDeviceCertLocation) {
		FUNC_EXIT_RC(NULL_VALUE_ERROR);
	}

	for(i = 0; i < AWS_IOT_MQTT_NUM_SUBSCRIBE_HANDLERS; ++i) {
		pClient->clientData.messageHandlers[i].topicName = NULL;
		pClient->clientData.messageHandlers[i].pApplicationHandler = NULL;
		pClient->clientData.messageHandlers[i].pApplicationHandlerData = NULL;
		pClient->clientData.messageHandlers[i].qos = QOS0;
	}

	pClient->clientData.commandTimeoutMs = pInitParams->mqttCommandTimeout_ms;
	pClient->clientData.writeBufSize = AWS_IOT_MQTT_TX_BUF_LEN;
	pClient->clientData.readBufSize = AWS_IOT_MQTT_RX_BUF_LEN;
	pClient->clientData.counterNetworkDisconnected = 0;
	pClient->clientData.disconnectHandler = pInitParams->disconnectHandler;
	pClient->clientData.disconnectHandlerData = pInitParams->disconnectHandlerData;
	pClient->clientData.nextPacketId = 1;

	/* Initialize default connection options */
	rc = aws_iot_mqtt_set_connect_params(pClient, &default_options);
	if(SUCCESS != rc) {
		FUNC_EXIT_RC(rc);
	}

#ifdef _ENABLE_THREAD_SUPPORT_
	pClient->clientData.isBlockOnThreadLockEnabled = pInitParams->isBlockOnThreadLockEnabled;
	rc = aws_iot_thread_mutex_init(&(pClient->clientData.state_change_mutex));
	if(SUCCESS != rc) {
		FUNC_EXIT_RC(rc);
	}
	rc = aws_iot_thread_mutex_init(&(pClient->clientData.tls_read_mutex));
	if(SUCCESS != rc) {
		FUNC_EXIT_RC(rc);
	}
	rc = aws_iot_thread_mutex_init(&(pClient->clientData.tls_write_mutex));
	if(SUCCESS != rc) {
		FUNC_EXIT_RC(rc);
	}
#endif

	pClient->clientStatus.isPingOutstanding = 0;
	pClient->clientStatus.isAutoReconnectEnabled = pInitParams->enableAutoReconnect;

	rc = iot_tls_init(&(pClient->networkStack), pInitParams->pRootCALocation, pInitParams->pDeviceCertLocation,
					  pInitParams->pDevicePrivateKeyLocation, pInitParams->pHostURL, pInitParams->port,
					  pInitParams->tlsHandshakeTimeout_ms, pInitParams->isSSLHostnameVerify);

	if(SUCCESS != rc) {
		pClient->clientStatus.clientState = CLIENT_STATE_INVALID;
		FUNC_EXIT_RC(rc);
	}

	init_timer(&(pClient->pingTimer));
	init_timer(&(pClient->reconnectDelayTimer));

	pClient->clientStatus.clientState = CLIENT_STATE_INITIALIZED;

	FUNC_EXIT_RC(SUCCESS);
}

uint16_t aws_iot_mqtt_get_next_packet_id(AWS_IoT_Client *pClient) {
	return pClient->clientData.nextPacketId = (uint16_t) ((MAX_PACKET_ID == pClient->clientData.nextPacketId) ? 1 : (
			pClient->clientData.nextPacketId + 1));
}

bool aws_iot_mqtt_is_client_connected(AWS_IoT_Client *pClient) {
	bool isConnected;

	FUNC_ENTRY;

	if(NULL == pClient) {
		IOT_WARN(" Client is null! ");
		FUNC_EXIT_RC(false);
	}

	switch(pClient->clientStatus.clientState) {
		case CLIENT_STATE_INVALID:
		case CLIENT_STATE_INITIALIZED:
		case CLIENT_STATE_CONNECTING:
			isConnected = false;
			break;
		case CLIENT_STATE_CONNECTED_IDLE:
		case CLIENT_STATE_CONNECTED_YIELD_IN_PROGRESS:
		case CLIENT_STATE_CONNECTED_PUBLISH_IN_PROGRESS:
		case CLIENT_STATE_CONNECTED_SUBSCRIBE_IN_PROGRESS:
		case CLIENT_STATE_CONNECTED_UNSUBSCRIBE_IN_PROGRESS:
		case CLIENT_STATE_CONNECTED_RESUBSCRIBE_IN_PROGRESS:
		case CLIENT_STATE_CONNECTED_WAIT_FOR_CB_RETURN:
			isConnected = true;
			break;
		case CLIENT_STATE_DISCONNECTING:
		case CLIENT_STATE_DISCONNECTED_ERROR:
		case CLIENT_STATE_DISCONNECTED_MANUALLY:
		case CLIENT_STATE_PENDING_RECONNECT:
		default:
			isConnected = false;
			break;
	}

	FUNC_EXIT_RC(isConnected);
}

bool aws_iot_is_autoreconnect_enabled(AWS_IoT_Client *pClient) {
	FUNC_ENTRY;
	if(NULL == pClient) {
		IOT_WARN(" Client is null! ");
		FUNC_EXIT_RC(false);
	}

	FUNC_EXIT_RC(pClient->clientStatus.isAutoReconnectEnabled);
}

IoT_Error_t aws_iot_mqtt_autoreconnect_set_status(AWS_IoT_Client *pClient, bool newStatus) {
	FUNC_ENTRY;
	if(NULL == pClient) {
		FUNC_EXIT_RC(NULL_VALUE_ERROR);
	}
	pClient->clientStatus.isAutoReconnectEnabled = newStatus;
	FUNC_EXIT_RC(SUCCESS);
}

IoT_Error_t aws_iot_mqtt_set_disconnect_handler(AWS_IoT_Client *pClient, iot_disconnect_handler pDisconnectHandler,
												void *pDisconnectHandlerData) {
	FUNC_ENTRY;
	if(NULL == pClient || NULL == pDisconnectHandler) {
		FUNC_EXIT_RC(NULL_VALUE_ERROR);
	}

	pClient->clientData.disconnectHandler = pDisconnectHandler;
	pClient->clientData.disconnectHandlerData = pDisconnectHandlerData;
	FUNC_EXIT_RC(SUCCESS);
}

uint32_t aws_iot_mqtt_get_network_disconnected_count(AWS_IoT_Client *pClient) {
	return pClient->clientData.counterNetworkDisconnected;
}

void aws_iot_mqtt_reset_network_disconnected_count(AWS_IoT_Client *pClient) {
	pClient->clientData.counterNetworkDisconnected = 0;
}

#ifdef __cplusplus
}
#endif


text:
/*
    Special thanks to Giovanni Di Sirio for teachings, his moral support and
    friendship. Note that some or every piece of this file could be part of
    the ChibiOS project that is intellectual property of Giovanni Di Sirio.
    Please refer to ChibiOS/RT license before use this file.

	For suggestion or Bug report - roccomarco.guglielmi@playembedded.org
 */

/**
 * @file    max7219.c
 * @brief   MAX7219 display driver module code.
 *
 * @addtogroup max7219
 * @{
 */

#include "ch.h"
#include "hal.h"

#include "max7219.h"

/*===========================================================================*/
/* Driver local definitions.                                                 */
/*===========================================================================*/

/*===========================================================================*/
/* Driver exported variables.                                                */
/*===========================================================================*/

/*===========================================================================*/
/* Driver local variables and types.                                         */
/*===========================================================================*/

/*===========================================================================*/
/* Driver local functions.                                                   */
/*===========================================================================*/

/*===========================================================================*/
/* Driver exported functions.                                                */
/*===========================================================================*/

/**
 * @brief   Reads a generic register value.
 * @pre     The SPI interface must be initialized and the driver started.
 *
 * @param[in] spip      pointer to the SPI interface
 * @param[in] adr       address number
 * @param[in] data      data value.
 */
void max7219WriteRegister(SPIDriver *spip, uint16_t adr, uint8_t data) {

  switch (adr) {
    default:
      return;
    case MAX7219_AD_DIGIT_0:
    case MAX7219_AD_DIGIT_1:
    case MAX7219_AD_DIGIT_2:
    case MAX7219_AD_DIGIT_3:
    case MAX7219_AD_DIGIT_4:
    case MAX7219_AD_DIGIT_5:
    case MAX7219_AD_DIGIT_6:
    case MAX7219_AD_DIGIT_7:
    case MAX7219_AD_DECODE_MODE:
    case MAX7219_AD_INTENSITY:
    case MAX7219_AD_SCAN_LIMIT:
    case MAX7219_AD_SHUTDOWN:
    case MAX7219_AD_DISPLAY_TEST:
      spiSelect(spip);
      uint16_t txbuf = {adr | data};
      spiSend(spip, 1, &txbuf);
      spiUnselect(spip);
  }
}
/** @} */

redpajama_set_name: RedPajamaGithub


Space Pirates And Zombies 2 Cheats, Codes, Hints and Walkthroughs for PC Games.
While playing the game, press [Alt] + [F7] to unlock the "Cheats" menu.
An Old Friend      : Meet someone from your past and blow up their stuff.
Basic Training     : Systems checks complete!
Disinfected        : Clear the galaxy of infection.
Go There, Do That! : Learn how to command strike craft.
Level 13           : The bane of Carl's existence.
Live Fire Exercise : Don't worry, it was just a bandit.
Matricide          : You destroyed 10 motherships.
Megacide           : You destroyed 100 motherships.
Raid Buster        : Blow up some defenseless strike craft.
Sticky Mess        : Investigate the zombie signal.
The End            : Complete the story.


I have been paddling for the last 11 years and during that time, I have been lucky enough to paddle in some of the coolest locations around the world from the UK to Chile, France, India and New Zealand. My personal favorite and the place that constantly draws me back is Norway. It's a land full of amazing rivers with still so much undiscovered potential. For the last 5 years I have spent my life working as a raft guide which allowed me to be closer to the river and to get as much boating in as possible. I also have the pleasure of being supported by some of the best companies on the market: Zet UK, AS watersports, System X and Silverstick.
Sweet Wanderer – Only the best to keep you head safe.
Palm FXr – Simple, light and does everything you need it to do.
IR Royale – It will not fail you when you need it, enough said.
Kokatat – They produce the best dry gear on the market and it is the only dry suit I have ever had that is truly a drysuit.
Werner Power House 200cm straight-shaft – It's reliable and powerful, exactly what what you need.
Zet Raptor – It's my main boat of choice. I am eager to get my hands on the new Toro as it looks amazing. I also have a Molan for a little bit of playboating and a old Riot Showbiz for some old school fun now and then.
I have a selection but manly anything merino wool as it doesn't smell as bad. Most of my merino wool is acquired from the Lost & Found at the rafting base.
I have a few upcoming plans for spring and summer 2015. I plan to be chasing the water in Europe in the spring with either a trip to Italy or in Telemark, Norway depending on where the water will be. I will be back in Voss for rafting season and plenty of fun. I will be going on a mid-summer trip to Northern Norway and followed by going to Mexico for a month in the autumn. It should be another amazing year.


LK 19:1 And Jesus entered and passed through Jericho.
LK 19:2 And, behold, there was a man named Zacchaeus, which was the chief among the publicans, and he was rich.
LK 19:3 And he sought to see Jesus who he was; and could not for the press, because he was little of stature.
LK 19:4 And he ran before, and climbed up into a sycomore tree to see him: for he was to pass that way.
LK 19:5 And when Jesus came to the place, he looked up, and saw him, and said unto him, Zacchaeus, make haste, and come down; for to day I must abide at thy house.
LK 19:6 And he made haste, and came down, and received him joyfully.
LK 19:7 And when they saw it, they all murmured, saying, That he was gone to be guest with a man that is a sinner.
LK 19:8 And Zacchaeus stood, and said unto the Lord: Behold, Lord, the half of my goods I give to the poor; and if I have taken any thing from any man by false accusation, I restore him fourfold.
LK 19:9 And Jesus said unto him, This day is salvation come to this house, forsomuch as he also is a son of Abraham.
LK 19:10 For the Son of man is come to seek and to save that which was lost.
LK 19:11 And as they heard these things, he added and spake a parable, because he was nigh to Jerusalem, and because they thought that the kingdom of God should immediately appear.
LK 19:12 He said therefore, A certain nobleman went into a far country to receive for himself a kingdom, and to return.
LK 19:13 And he called his ten servants, and delivered them ten pounds, and said unto them, Occupy till I come.
LK 19:14 But his citizens hated him, and sent a message after him, saying, We will not have this man to reign over us.
LK 19:15 And it came to pass, that when he was returned, having received the kingdom, then he commanded these servants to be called unto him, to whom he had given the money, that he might know how much every man had gained by trading.
LK 19:16 Then came the first, saying, Lord, thy pound hath gained ten pounds.
LK 19:17 And he said unto him, Well, thou good servant: because thou hast been faithful in a very little, have thou authority over ten cities.
LK 19:18 And the second came, saying, Lord, thy pound hath gained five pounds.
LK 19:19 And he said likewise to him, Be thou also over five cities.
LK 19:21 For I feared thee, because thou art an austere man: thou takest up that thou layedst not down, and reapest that thou didst not sow.
LK 19:23 Wherefore then gavest not thou my money into the bank, that at my coming I might have required mine own with usury?
LK 19:24 And he said unto them that stood by, Take from him the pound, and give it to him that hath ten pounds.
LK 19:26 For I say unto you, That unto every one which hath shall be given; and from him that hath not, even that he hath shall be taken away from him.
LK 19:27 But those mine enemies, which would not that I should reign over them, bring hither, and slay them before me.
LK 19:28 And when he had thus spoken, he went before, ascending up to Jerusalem.
LK 19:30 Saying, Go ye into the village over against you; in the which at your entering ye shall find a colt tied, whereon yet never man sat: loose him, and bring him hither.
LK 19:31 And if any man ask you, Why do ye loose him? thus shall ye say unto him, Because the Lord hath need of him.
LK 19:32 And they that were sent went their way, and found even as he had said unto them.
LK 19:33 And as they were loosing the colt, the owners thereof said unto them, Why loose ye the colt?
LK 19:34 And they said, The Lord hath need of him.
LK 19:35 And they brought him to Jesus: and they cast their garments upon the colt, and they set Jesus thereon.
LK 19:36 And as he went, they spread their clothes in the way.
LK 19:38 Saying, Blessed be the King that cometh in the name of the Lord: peace in heaven, and glory in the highest.
LK 19:39 And some of the Pharisees from among the multitude said unto him, Master, rebuke thy disciples.
LK 19:40 And he answered and said unto them, I tell you that, if these should hold their peace, the stones would immediately cry out.
LK 19:42 Saying, If thou hadst known, even thou, at least in this thy day, the things which belong unto thy peace! but now they are hid from thine eyes.
LK 19:44 And shall lay thee even with the ground, and thy children within thee; and they shall not leave in thee one stone upon another; because thou knewest not the time of thy visitation.
LK 19:46 Saying unto them, It is written, My house is the house of prayer: but ye have made it a den of thieves.
LK 19:48 And could not find what they might do: for all the people were very attentive to hear him.


Amazingly, soon after blogging my grumble about the frustrated imp running round in my head and making a racket, there was enough of a lull in the wind for me to finally get Bob back on board. And once again, a beautiful Sooty albatross appeared at just the right time, dissolving any trace of grumbles from before. I love it how they do that. Albie total now eleven!
So we have just had two exhilarating hours surfing down and across some enormous waves before dark crept up on us. I shall leave Dippers to it overnight – she surfs like a dream and with thick cloud cover it's not very safe or pleasant to row in big seas in the pitch black. Oars in face, big waves over your head etc.
Now we shall have real smiles (as opposed to my forced one from earlier) for Day 60. *Smiles real smiles* We are safe, we are happy and we are going West on our new Red Carpet. And tomorrow we have a whole day of surfing. Wohoooo!
Have yourselves a happy weekend (I can recommend albatross and chocolate biscuit therapy if happyometer is flagging at half mast) and I'll touch base again in due course. Due West!
Monday will herald the start of month three at sea. Month Three.
many miles West as we did in just two and a half days during our last run on the Red Carpet.
Now where would be the fun if it were plain sailing\rowing all the way? Glad Bob is now in his rightful place, may your red carpet stretch all the way to the end. I think you have proved you can deal with the tough times, now for some surfing!
Brilliant news, Sarah. Happy surfing along the red carpet!
Please send it Ferkins way ….
Catch a wave and you'r on the top of the world. Good surfing to you now that the wind and sea anchor are all in their proper places.I see naughties in your future!
And he said when you get back, that he will get in touch because he would like to make a big donation personally!! How lovely and thoughtful is that?
What a nice, talented, down to earth man.
Hope this news puts a smile on your face, and cheers your day after your rough ride!
Good to hear all yout news and that Bob is behaving himslef again now, Happy surfing for many miles, we are all behind you. You will soon have reached your target and many more I should think, the followers are growing.
We all know you have some good ideas!!!
Ah they do have a musical candle they can light and sing to.It came in handy three days into the race when Guy had his birthday.
Girl power to the rescue.
"Happy Socks and the Green Line" – sounds like a good children's title ?
Hi Outey, great pics of the millpond. I think I was watching those wonderful Albies fairly close to your @ when I was 24.
Not good when you are unable to recover Bob and rather than helping, he is slowing your progress and causing other grief. Perhaps you should train the Tweedles to nibble him into submission.
Perhaps he has ideas above his station as the Royal Bob ? So, have you now worked out how to rig a trip line to his crown ?
Happy Westward surfing and hang Ten, but don't wipeout by playing submarines !
Sarah: Perhaps in future versions of BOB a retrieval line might be connected to BOB's backside so that he turns around when being hauled in. Not sure what you might have on hand to make that change but it might be worth a try.
Ian, I think that if you want to use photos from the gallery for press etc it is hoped that you would credit them accordingly……see note on Media page of website. If you need to know any more, contact Amy at Whisper P R .
If you get blasé with the Red Carpet, let me know. I'll leave the one I am planning to roll out for you in Mauritius, in the warehouse.
Wishing you safe and fast winds and many more Albatross.
The Mauritius Wildlife Foundation would like you to visit Ile aux Aigrettes. A little islet of some 20 hectares, in Vieux Grand Port, that they have rid of all non-endemic flora and fauna…in 15 years of hard work. My old friend, Welshman Prof.Carl Jones started it all and another old friend Tim Taylor (Scottish parents) is the new Chairman.


text:
\section{Introduction}

The coarse geometric Novikov conjecture provides an algorithm of determining non-vanishing of the higher index for elliptic differential operators on noncompact Riemannian manifolds. It implies Gromov's conjecture stating that a uniformly contractible Riemannian manifold with bounded geometry cannot have uniformly positive scalar curvature, and the zero-in-the-spectrum conjecture stating that the Laplacian operator acting on the space of all $L^2$-forms of a uniformly contractible Riemannian manifold has zero in its spectrum [9,13,17,22-30]

Recently, an $\lp$ analog of the coarse geometric Novikov conjecture for $1<p<\infty$ was introduced in \cite{CN}. Although the $\lp$ analog of the coarse Novikov conjecture has no known geometric or topological applications when $p\neq 2$, this study contributes to the general interests of understanding of the $K$-theory of some operator algebras. For example, Zhang and Zhou proved that $K$-theory for $L^p$ Roe algebra of a finite asymptotic dimensional metric space does not depend on $p$ in \cite{ZZ}. More related references are included in \cite{CN}.

In this paper, we prove the following result.
\begin{thm}\label{main}
Let $\Gamma$ be a discrete metric space with bounded geometry. If $\Gamma$ admits a coarse embedding into a simply-connected complete Riemannian manifold of non-positive sectional curvature, then the coarse geometric $\ell^p$-Novikov conjecture holds for $\Gamma$, i.e., the index map
$$
e_*:\lim_{d\to\infty}K_*(\clp(\pdg))\to K_*(\cp(\Gamma))
$$
is injective.
\end{thm}

Recall that for two metric spaces $X$ and $Y$, a map $f:X\to Y$ is said to be a {\it coarse embedding} \cite{Grom} if there exist non-decreasing functions $\rho_1$ and $\rho_2$ from $\mathbb R_+=[0,\infty)$ to $\mathbb R_+$ such that:
\begin{enumerate}
\item $\rho_1(d(x,y))\leq d(f(x),f(y))\leq \rho_2(d(x,y))$;
\item $\lim_{r\to\infty}\rho_i(r)=\infty$ for $i=1,2$.
\end{enumerate}

\section{The coarse geometric Novikov conjecture}
In this section, we shall recall the concepts of the $\lp$-Roe algebra \cite{Roe93}, Yu's $\lp$-localization algebras \cite{Yu97} and the coarse geometric $\ell^p$-Novikov conjecture.

Let $X$ be a metric space. $X$ is called $proper$ if every closed ball is compact. When $X$ is discrete, $X$ has $bounded\ geometry$ if for any $R>0$, there exists $M_R>0$ such that for any $x\in X$ the cardinality $|B(x; R)|$ is less than or equal to $M_R$. For $r>0$, a $r$-net in $X$ is a discrete subset $Y\subset X$ such that for any $y_1,y_2\in Y$, $d(y_1,y_2)\geq r $ and for any $x\in X$ there is a $y\in Y$ such that $d(x,y)<r$. A general metric space $X$ is called to have bounded geometry if $X$ has a $r$-net $Y$ for some $r>0$ such that $Y$ has bounded geometry.

Throughout the paper, $p>1$. And $\calk_p=\calk(\ell^p)$, the set of all compact operators over $\ell^p$.

\begin{defn}[\cite{Roe93,CN}]
Let $X$ be a proper metric space, and fix a countable dense subset $Z\subseteq X$. Let $T$ be a bounded operator on $\lp(Z,\lp)$, and write $T=(T(x,y))_{x,y\in Z}$ so that each $T(x,y)$ is a bounded operator on $\lp$. $T$ is said to be {\it locally compact} if
\begin{itemize}
\item each $T(x,y)$ is a compact operator on $\lp$;
\item for every bounded subset $B\subseteq X$, the set
$$
\set{(x,y)\in(B\times B)\cap(Z\times Z):T(x,y)\neq 0}
$$
is finite.
\end{itemize}
The {\it propagation} of $T$ is defined to be
$$
propagation(T)=\inf\set{S>0:T(x,y)=0\text{ for all }x,y\in Z\text{ with }d(x,y)>S}.
$$
The {\it algebraic $\lp$ Roe algebra} of $X$, denoted by $\calgp(X)$, is the subalgebra of $\call(\lp(Z,\lp))$ consisting of all finite propagation, locally compact operators. The {\it $\lp$ Roe algebra} of $X$, denoted by $\cp(X)$, is the closure of $\calgp(X)$ in $\call(\lp(Z,\lp))$.
\end{defn}

$\cp(X)$ does not depend on the choice of $Z$. See \cite{ZZ} for a proof.

\begin{defn}[\cite{Yu97}]  The {\em $\lp$-localization algebra} $\clp(X)$ is the norm-closure of the algebra of all
bounded and uniformly norm-continuous functions $g: \; [0,\infty)\to \cp(X)$ such that
$$
propagation(g(t))\to 0\quad\mbox{as}\quad t\to \infty.
$$
\end{defn}

The evaluation homomorphism $e$ from $\clp(X)$ to $\cp(X)$ is defined by $e(g)=g(0)$ for $g\in \clp(X)$.

\begin{defn}
Let $\Gamma$ be a discrete metric space and $d\geq 0$. The Rips complex of $\Gamma$ at scale $d$, denoted by $P_d(\Gamma)$, is the simplicial complex with vertex set $\Gamma$, and where a finite subset $E\subset \Gamma$ spans a simplex if and only if $d(g,h)\geq d$ for all $g,h\in E$.

Points in $P_d(\Gamma)$ can be written as formal linear combinations $\sum_{g\in\Gamma}t_gg$, where $t_g\in [0,1]$ for each $g$ and $\sum_{g\in \Gamma}t_g=1$. $P_d(\Gamma)$ is equipped with the $\ell^1$ metric, i.e.,
$$
d(\sum_{g\in\Gamma}t_gg,\sum_{g\in\Gamma}s_gg)=\sum_{g\in\Gamma}|t_g-s_g|.
$$
\end{defn}

To define the assembly map, we recall that when $p=2$, Yu in \cite{Yu97} proved that the local index map from $K$-homology to $K$-theory of localization algebra is an isomorphism for finite-dimensional simplicial complexes. Qiao and Roe in \cite{QR} later generalized this isomorphism to general locally compact metric spaces. Therefore for $p\in(1,\infty)$, considering the analogs of $\lp$-Roe algebra and $\lp$-localization algebra, we get the following assembly map which is equivalent to the original map when $p=2$. The following conjecture is called the coarse geometric $\ell^p$-Novikov conjecture:

\begin{conj} If $\Gamma$ is a discrete metric space with bounded geometry, then the index map
$$
e_*: \lim_{d\to \infty} K_*(\clp(P_d(\Gamma))) \to \lim_{d\to\infty} K_*(\cp(P_d(\Gamma)))\cong
K_*(\cp(\Gamma))
$$
is injective.
\end{conj}

\section{An $\ell^p$ coarse Mayer-Vietoris principle}
In this section, we present an $\ell^p$ coarse Mayer-Vietoris principle similar to the argument in \cite{HRY}.

\begin{defn}[\cite{HRY}]
Let $X$ be a proper metric space, and let $A$ and $B$ be closed subspace with $X=A\cup B$. We say that $(A,B)$ is a $w$-excisive couple, or that $X=A\cup B$ is a $w$-excisive decomposition, if for each $R>0$ there is some $S>0$ such that
$$
Pen(A;R)\cap Pen(B;R)\subset Pen(A\cap B;S),
$$
where $Pen(A;R)$ is the set of points in $X$ of distance at most $R$ from $A$.
\end{defn}

\begin{defn}[\cite{HRY}]
Let $A$ be a closed subspace of a proper metric space $X$. Denote by $\cp(A;X)$ the operator-norm closure of the set of all locally compact, finite propagation operators $T$ whose support is contained in $Pen(A;R)\times Pen(A;R)$, for some $R>0$ (depending on $T$).
\end{defn}

One can see that $\cp(A;X)$ is a two-sided ideal of $\cp(X)$. For $s,t\in [0,\infty)$ with $s<t$, let $i_{s,t}:Pen(A;s)\to Pen(A;t)$ be the inclusion map. Then $i_{s,t}$ induces a map
$$
i_{s,t}^*:\ell^p(Pen(A;t),\calk_p)\to \ell^p(Pen(A;s),\calk_p).
$$
Then the induced map $i_{s,t,*}:\cp(Pen(A;s))\to \cp(Pen(A;t))$ is defined by, for any $f\in \ell^p(Pen(A;t),\calk_p)$ and $T\in \cp(Pen(A;s))$,
$$
i_{s,t,*}(T)(f)=T(i_{s,t}^*(f)).
$$
By definition, we have $i_{t,r,*}\circ i_{s,t,*}=i_{s,r,*}$ for any $s<t<r$. And $\norm{i_{s,t,*}(T)}_t\leq\norm{T}_s$ for $T\in \cp(Pen(A;s))$. Here we identify $A$ as $Pen(A;0)$. Since $\cp(A;X)=\lim_{n\to\infty} \cp(Pen(A;n))$, we define that, for any $T\in \cp(A)$, $i_*:K_*(\cp(A))\to K_*(\cp(A;X))$ by
$$
i_*(T)=\lim_{n\to\infty}i_{0,n,*}(T).
$$
\begin{lem}[\cite{HRY}]
The induced map
$$
i_*:K_*(\cp(A))\to K_*(\cp(A;X))
$$
is an isomorphism.
\end{lem}
\begin{proof}
This is because that the inclusions $A\subset Pen(A;n)$ and $Pen(A;n)\subset Pen(A;n+1)$ are coarsely equivalent, hence the induced maps on $K$-theory are all isomorphisms.
\end{proof}

Let $X=A\cup B$. Let $I=\cp(A;X)$ and $J=\cp(B;X)$. Define $U:A\times A\to \calk_p$ such that
\begin{align*}
&U(x,y)=0\text{ if } x\neq y, \\
&U(x,x)=\left(\begin{array}{cc}
                   I_{r(x)} & 0 \\
                   0 & 0 \\
                 \end{array}
               \right),
\end{align*}
$\text{where }I_{r(x)}\text{ is a rank }r(x)\text{ identity matrix for some }r(x)\in\bbn.$
We define a partial order on all such $U$ by the following: $U_2\leq U_1$ if
$
\rank(U_2(x,x))\leq \rank(U_1(x,x))$ for all $x\in X.$ Let $\calu$ be the set of all such operators $U$ with this order.

\begin{prop}\label{approximateunit}
The collection $\calu$ is an approximate unit of $I$.
\end{prop}
\begin{proof}
Let $T\in I$, for any $\epsilon>0$, for any $x,y\in X$, $T(x,y)$ is either a zero operator or a compact operator over $\ell^p$. $X\times X$ is countable, so each pair $(x,y)\in X\times X$ has a corresponding integer $n$.

Let $F(x,y)$ be a finite rank operator over $\ell^p$ such that $\norm{T(x,y)-F(x,y)}<\frac{1}{2^n}\epsilon$ when $T(x,y)\neq 0$ and $n$ is the corresponding integer of $(x,y)$, and $F(x,y)=0$ when $T(x,y)=0$. Then $F=\left(F(x,y)\right)$ is a locally  finite rank operator of finite propagation with $\norm{T-F}<\epsilon$.

For each fixed $x$, since $F$ has finite propagation, there are only finitely many $y$ such that $F(x,y)\neq 0$. Let $U(x,x)=\left(\begin{array}{cc}
                   I_{r(x)} & 0 \\
                   0 & 0 \\
                 \end{array}\right)$ be a finite-rank projection for some $r(x)\in\bbn$ such that $U(x,x)F(x,y)=F(x,y)$ for all $y$ with $F(x,y)\neq 0$. Then define a $U_\lambda=\set{
U(x,y)=0\text{ if } x\neq y, \text{ and }
U(x,x)=\left(\begin{array}{cc}
                   I_{r(x)} & 0 \\
                   0 & 0 \\
                 \end{array}
               \right)}$ where $U(x,x)$ is defined ahead.

Then
$$
\norm{U_\lambda T-T}\leq \norm{U_\lambda T-U_\lambda F}+\norm{U_\lambda F-F}+\norm{F-T}.
$$
Here
$\norm{F-T}\leq \epsilon$, $U_\lambda F-F=0$ and $\norm{U_\lambda T-U_\lambda F}\leq\norm{U_\lambda}\norm{F-T}<\epsilon$. So $\norm{U_\lambda T-T}\leq 2\epsilon$ and the proof is done.
\end{proof}

\begin{prop}\hfill
\begin{enumerate}
\item $\cp(A;X)+\cp(B;X)=\cp(X)$ for any decomposition $A$ and $B$ of $X$;
\item $\cp(A;X)\cap \cp(B;X)=\cp(A\cap B;X)$ if $A$ and $B$ are $w$-excisive.
\end{enumerate}
\end{prop}
\begin{proof}
Naturally, $\cp(A;X)+\cp(B;X)\subset\cp(X)$. For any $T\in\calgp(X)$, assume that $propagation(T)=R$. Let $\chi_A$ be the characteristic function of $A$. Then $T\cdot\chi_A\in \cp(A;X)$ and $T\cdot(1-\chi_A)\in\cp(B;X)$, and $T=T\chi_A+T(1-\chi_A)\in\cp(A;X)+\cp(B;X)$.  Hence $\cp(A;X)+\cp(B;X)=\cp(X)$.

For the second part, $\cp(A\cap B;X)\subset \cp(A;X)\cap \cp(B;X)$ holds for any decomposition pair $(A,B)$. By Proposition \ref{approximateunit}, one can easily see that $\cp(A;X)\cap \cp(B;X)=\cp(A;X)\cp(B;X)$. For $T_A\in\calgp(A;X)$ and $T_B\in \calgp(B;X)$ with
\begin{align*}
Supp(T_A)&\subset Pen(A;R')\times Pen(A;R');\\
Supp(T_B)&\subset Pen(B;R'')\times Pen(B;R'').
\end{align*}
$(A,B)$ is $w$-excisive, then there exists $S>0$ such that
$$
Supp(T_AT_B)\subset Pen(A\cap B;S)\times Pen(A\cap B;S).
$$
Hence $\cp(A;X)\cp(B;X)\subset \cp(A\cap B;X)$.
\end{proof}
Combining these lemmas, we have the following $\lp$ coarse Mayer-Vietoris principle.
\begin{prop}
Let $A$ and $B$ be a $w$-excisive decomposition of $X$, then the following sequence is exact:
\begin{align*}
\cdots\to  K_j(\cp(A\cap &B)\to K_j(\cp(A))\oplus K_j(\cp(B))\\
&\to K_j(\cp(X))\to K_{j-1}(\cp(A\cap B))\to\cdots
\end{align*}
\end{prop}

\section{Twisted $\lp$-Roe algebras and twisted $\lp$-localization algebras}
In this section, we shall define the twisted $\lp$-Roe algebras and the twisted $\lp$-localization algebras for bounded
geometry spaces which admit a coarse embedding into a simply connected complete Riemannian manifold of
nonpositive sectional curvature. The construction of these twisted $\lp$-algebras is similar to those twisted algebras
introduced in \cite{Yu00}.

Let $M$ be a simply connected complete Riemannian manifold of nonpositive sectional curvature. In the following,
we shall assume that the dimension of $M$ is even. If $\dim(M)$ is odd, we can replace $M$ by $M\times \bbr$.
Indeed, the product manifold $M\times \bbr$ is also a simply connected complete Riemannian manifold
with nonpositive sectional curvature. And if $f: \Gamma\to M$ is a coarse embedding, then the induced map
$f': \Gamma\to M\times \bbr$ defined by $f'(\gamma)=(f(\gamma), 0)$ is also a coarse embedding so that
we can replace $f$ by $f'$. Thus, without loss of generality, we assume $\dim M=2n$ for some integer $n>0$.

Let $\cA=C_0(M,\cliff_{\bbc}(TM))$ be the $C^*$-algebra of continuous functions $a$ on $M$ which have value $a(x)\in \cliff_\bbc(T_xM)$ at each point $x\in M$ and vanish at infinity, where $\cliff_\bbc(T_xM)$ is the complexified Clifford algebra \cite{ABS,LM} of the tangent space $T_xM$ at $x\in M$ with respect to the inner product on $T_xM$ given by the Riemannian structure of $M$. Then $\cliff_{\bbc}(TM)$ is the Clifford bundle over $M$. Meanwhile, for any $x\in M$, $\cliff_\bbc(T_xM)$ is also a Hilbert space, so that $\cliff_\bbc(TM)$ ia also a Hilbert space bundle. Let $\calh=L^2(M,\cliff_{\bbc}(TM))$, the set of all $L^2$ sections of $\cliff_\bbc(TM)$, which is a Hilbert space. $\cala$ acts on $\calh$ by pointwise multiplication. For $a\in \cala$ and $h\in\calh$. Define $a_{\max}=\max\set{\norm{a(x)}|x\in M}$. Then $\norm{a\cdot h}\leq a_{\max}\norm{h}$ and $\cala\subset \call(\calh)$. For $n\in\bbn$, define $\calh_{n,p}=\calh\oplus_p\cdots\oplus_p \calh$, the $\lp$-direct sum of $n$ copies of $\calh$. The $\lp$-norm of $\calh_{n,p}$ is defined as
$$
\norm{(f_1,\cdots,f_n)}_{p}=\sqrt[p]{\sum^n_{i=1}\norm{f_i}^p},\quad\quad \text{for }f_1,\cdots,f_n\in \calh.
$$
Let $M_n(\cala)$ be the set of $n\times n$ matrices with entries in $\cala$. Then elements of $M_n(\cala)$ act on $\calh_{n,p}$ by matrix multiplication. For $a=(a_{i,j})_{i,j\in\set{1,\cdots,n}}\in M_n(\cala)$ and $h_n\in \calh_{n,p}$, $\norm{a\cdot h_n}\leq \max_{i,j\in\set{1,\cdots,n}}\set{\left(a_{i,j}\right)_{\max}}\cdot\norm{h_n}$. Hence $M_n(\cala)\subset \call(\calh_{n,p})$. Let $r_{n,n+1}:\calh_{n+1,p}\to \calh_{n,p}$ be the projection map defined by
$$
r_{n,n+1}(h_1,\cdots,h_n,h_{n+1})=(h_1,\cdots,h_n),
$$
for $(h_1,\cdots,h_{n+1})\in\calh_{n,p}$. Then $r_{n,n+1}^*(M_n(\cala))\subset M_{n+1}(\cala)$. This is equivalent to embed $M_n(\cala)$ into $M_{n+1}(\cala)$ by placing matrices at the top left corner and inserting 0 at the right column and the bottom line. And $\norm{r_{n,n+1}^*(M)}\leq\norm{M}$ for all $M\in M_n(\cala)$. Let $M_{\infty,p}(\cala)$ be the inductive limit of $\set{M_{n}(\cala)}_{n=1}^\infty$. Define $\calh_{\infty,p}$ to be the $\lp$-direct sum of infinite copies of $\calh$ with the $\lp$-norm
$$
\norm{\set{f_i}^\infty_{i=1}}_p=\sqrt[p]{\sum^\infty_{i=1}\norm{f_i}^p},\quad\quad \text{for }\set{f_i}^\infty_{i=1}\in\calh_{\infty,p}.
$$
Then $\calh_{\infty,p}\cong \lp(\bbn,\calh)\cong\lp(\bbn)\otimes_p \calh$ and all $M_{n}(\cala)$ can be considered as subalgebras of $\call(\calh_{\infty,p})$. Denote by $\calk_p\otimes_{alg}\cala$ the algebraic tensor product of $\calk_p$ and $\cala$. Naturally $\calk_p\otimes_{alg}\cala$ acts on $\calh_{\infty,p}$ and $\calk_p\otimes_{alg}\cala\subset\call(\calh_{\infty,p})$. Let $\calk_p\otimes_p \cala=\overline{\calk_p\otimes_{alg}\cala}^{\call(\calh_{\infty,p})}$. It follows that $\calk_p\otimes_p \cala\cong M_\infty(\cala)$.

Let $\Gamma$ be a discrete metric space with bounded geometry. Let $f:\; \Gamma\to M$ be a coarse embedding.

For each $d> 0$, we shall extend the map $f$ to the Rips complex $P_d(\Gamma)$ in the following way.
Note that $f$ is a coarse map, i.e., there exists $R>0$ such that for all $\gamma_1,\gamma_2\in\Gamma$,
$$
d(\gamma_1,\gamma_2)\leq d \; \Longrightarrow  \; d_M(f(\gamma_1),f(\gamma_2))\leq R.
$$
For any point $x=\sum_{\gamma\in \Gamma} c_\gamma \gamma \in P_d(\Gamma)$, where  $c_\gamma\geq 0$ and
$\sum_{\gamma\in \Gamma} c_\gamma=1$,  we  choose a point $f_x\in M$ such that
$$
d(f_x, f(\gamma))\leq R
$$
for all $\gamma\in \Gamma$ with $c_\gamma\not=0$. The correspondence $x\mapsto f_x$ gives a coarse embedding
$P_d(\Gamma)\to M$, also denoted by $f$.

Choose a countable dense subset $\Gamma_d$ of $P_d(\Gamma)$ for each $d>0$ in such a way that
$\Gamma_d\subset \Gamma_{d'}$ when $d<d'$.

Let $\calgp(P_d(\Gamma),\cA)$ be the set of all functions
$$
T:\Gamma_d\times \Gamma_d\to \cK_p\otimes_p\cA\subset\call(\lp\otimes_p L^2(M,\cliff_\bbc(TM)))
$$
such that
\begin{enumerate}
\item there exists $C>0$ such that $\|T(x,y)\| \leq C$ for all $x,y\in \Gamma_d$;
\item there exists $R>0$ such that $T(x,y)=0$ if $d(x,y)>R$;
\item there exists $L>0$ such that for every $z\in P_d(\Gamma)$, the number of elements
in the following set
$$\{ (x, y)\in\Gamma_d\times \Gamma_d: \; d(x, z)\leq 3R, \; d(y, z)\leq 3R, \; T(x,y)\neq 0 \}$$
is less than $L$.
\item there exists $r>0$ such that
$$
\supp(T(x,y))\subset B(f(x),r)
$$
for all $x,y\in \Gamma_d$,  where $B(f(x),r)=\{m\in M: \; d(m,f(x))<r\}$ and, for all $x,y\in\Gamma_d$, the entry
$T(x,y)\in \cK_p\otimes_p\cA$ is a function on $M$ with $T(x, y)(m)\in \cK_p\otimes_p\Cliff_{\bbc}(T_mM)$ for each $m\in M$
so that the {\em support} of $T(x,y)$ is defined by
$$
\supp(T(x,y)):= \{m\in M: \; T(x,y)(m)\neq 0\}.
$$
\end{enumerate}

For $f\in \ell^p(\Gamma_d, \calhp)$, we define
$$
Tf(x)=\sum_{y\in\Gamma_d} T(x,y)f(y).
$$
Then $T=(T(x,y))\in \call(\ell^p(\Gamma_d,\calhp))$.

\begin{defn} The twisted $\lp$-Roe algebra $\cp(P_d(\Gamma),\cA)$ is defined to be the operator norm closure
of $\calgp(P_d(\Gamma), \cA)$ in $\call(\ell^p(\Gamma_d,\calhp))$.
\end{defn}

The above definition of the twisted $\lp$-Roe algebra is similar to that in \cite{Yu00}.

Let $\clalgp(P_d(\Gamma),\cA)$ be the set of all bounded, uniformly norm-continuous  functions
$$
g:\; \mathbb{R}_+\to \calgp(P_d(\Gamma),\cA)
$$
such that
\begin{enumerate}
\item there exists a bounded function $R(t): \;\mathbb{R}_+\to\mathbb{R}_+$ with
                $\displaystyle \lim_{t\to\infty} R(t)=0$ such that $(g(t))(x,y)=0$ whenever $d(x,y)>R(t)$;
\item there exists $L>0$ such that for every $z\in P_d(\Gamma)$, the number of elements
                    in the following set
                    $$\{ (x, y)\in\Gamma_d\times \Gamma_d: \; d(x, z)\leq 3R, \; d(y, z)\leq 3R, \; g(t)(x,y)\neq 0 \}$$
                    is less than $L$ for every $t\in \mathbb{R}_+$.
\item there exists $r>0$ such that $\supp((g(t))(x,y))\subset B(f(x),r)$ for all
                $t\in\mathbb{R}_+$, $x,y\in\Gamma_d$, where $f:P_d(\Gamma)\to M$ is the extension of the coarse
                embedding $f:\Gamma\to M$ and $B(f(x),r)=\{m\in M: d(m,f(x))<r\}$.
\end{enumerate}

\begin{defn} The twisted $\lp$-localization algebra $\clp(P_d(\Gamma),\cA)$ is defined to be the norm
completion of $\clalgp(\pdg,\cA)$, where $\clalgp(\pdg,\cA)$ is endowed with the norm
$$
\|g\|_\infty=\sup_{t\in\mathbb{R}_+} \|{g(t)}\|_{\cp(\pdg,\cA)}.
$$
\end{defn}

The above definition of the twisted $\lp$-localization Roe algebra is similar to that in \cite{Yu00}. The evaluation homomorphism $e$ from $\clp(\pdg,\cA)$ to $\cp(\pdg,\cA)$ defined by $e(g)=g(0)$ induces a homomorphism
at $K$-theory level:
$$
e_*: \lim_{d\to\infty}K_*(\clp(\pdg,\cA))\to\lim_{d\to\infty}K_*(\cp(\pdg,\cA)).
$$

\begin{thm}\label{twisted-iso}
Let $\Gamma$ be a discrete metric space with bounded geometry which admits a coarse embedding
$f: \Gamma\to M$ into a simply connected, complete Riemannian manifold $M$ of non-positive sectional curvature.
Then the homomorphism
$$
e_*: \lim_{d\to\infty}K_*(\clp(\pdg,\cA))\to\lim_{d\to\infty}K_*(\cp(\pdg,\cA)).
$$
is an isomorphism.
\end{thm}
The proof of Theorem \ref{twisted-iso} is similar to the proof of Theorem 6.8 in \cite{Yu00}. To begin with, we
need to discuss ideals of the twisted algebras associated to open subsets of the manifold $M$.

\begin{defn}\hfill
\begin{enumerate}
\item The $support$ of an element $T$ in $\calgp(\pdg,\cA)$ is defined to be
$$
\begin{array}{ccl}
\supp(T)  & = & \Big\{(x,y,m)\in\Gamma_d\times\Gamma_d\times M:\; m\in\supp(T(x,y))\Big\}    \\
          & = & \Big\{(x,y,m)\in\Gamma_d\times\Gamma_d\times M:\; (T(x,y))(m)\not=0 \Big\};
\end{array}
$$
\item The $support$ of an element $g$ in $\clalgp(\pdg,\cA)$ is defined to be
$$
\bigcup_{t\in\mathbb{R}_+} \supp(g(t)).
$$
\end{enumerate}
\end{defn}

Let $O\subset M$ be an open subset of $M$. Define $\calgp(\pdg,\cA)_O$ to be the subalgebra of
$\calgp(\pdg,\cA)$ consisting of all elements whose supports are contained in $\Gamma_d\times\Gamma_d\times O$,
i.e.,
$$
\calgp(\pdg,\cA)_O \; = \; \{T\in \calgp(\pdg,\cA): \; \supp(T(x,y))\subset O,\; \forall \; x,y\in\Gamma_d \}.
$$
Define $\cp(\pdg,\cA)_O$ to be the norm closure of $\calgp(\pdg,\cA)_O$.  Similarly, let
$$
\clalgp(\pdg,\cA)_O=\Big\{ g\in \clalgp(\pdg,\cA): \; \supp(g)\subset\Gamma_d\times\Gamma_d\times O \Big\}
$$
and define $\clp(\pdg,\cA)_O$ to be the norm closure of $\clalgp(\pdg,\cA)_O$ under the norm
$\|g\|_\infty=\sup_{t\in\mathbb{R}_+} \|{g(t)}\|_{\cp(\pdg,\cA)}$.

Note that $\cp(\pdg,\cA)_O$ and $\clp(\pdg,\cA)_O$ are closed two-sided ideals of $\cp(\pdg,\cA)$ and
$\clp(\pdg,\cA)$, respectively. We also have an evaluation homomorphism
$e: \clp(\pdg,\cA)_O \to \cp(\pdg,\cA)_O$ given by $e(g)=g(0)$.

\begin{lem}\label{decomposition}
For any two open subsets $O_1, O_2$ of $M$, we have
$$\cspace_{O_1}+\cspace_{O_2}=\cspace_{O_1\cup O_2},$$
$$\cspace_{O_1}\cap\cspace_{O_2}=\cspace_{O_1\cap O_2},$$
$$\clspace_{O_1}+\clspace_{O_2}=\clspace_{O_1\cup O_2},$$
$$\clspace_{O_1}\cap\clspace_{O_2}=\clspace_{O_1\cap O_2}.$$
Consequently,  we have the following commuting diagram connecting two Mayer-Vietoris sequences at $K$-Theory level:
\[
\xymatrix
{
  & AL_0 \ar[rr] \ar'[d][dd]     &  &  BL_0\ar[rr]\ar'[d][dd]  &  &  CL_0 \ar[dd]^{e_*} \ar[dl]        \\
    CL_1 \ar[ur]\ar[dd]_{e_*}    &  &  BL_1 \ar[ll]\ar[dd]     &  &  AL_1 \ar[ll]\ar[dd]          &    \\
  & A_0 \ar'[r][rr]              &  &  B_0 \ar'[r][rr]         &  &  C_0 \ar[dl]                       \\
    C_1 \ar[ur]                  &  &  B_1\ar[ll]              &  &  A_1 \ar[ll]                  &        }
\]
where, for $*=0, 1$,
$$AL_*=K_*\Big(\clspace_{O_1\cap O_2}\Big), \quad CL_*=K_*\Big(\clspace_{O_1\cup O_2}\Big),$$
$$\;\;\; A_*=K_*\Big(\cspace_{O_1\cap O_2}\Big), \quad \;\; C_*=K_*\Big(\cspace_{O_1\cup O_2}\Big),$$
$$BL_*=K_*\Big(\clspace_{O_1}\Big)\bigoplus K_*\Big(\clspace_{O_2}\Big),$$
$$\;\; \; B_*=K_*\Big(\cspace_{O_1}\Big)\bigoplus K_*\Big(\cspace_{O_2}\Big).
$$
\end{lem}
\begin{proof}We shall prove the first equality. Other equalities can be proved similarly. Then the two
Mayer-Vietoris exact sequences follow from Lemma 2.4 of \cite{HRY}.

To prove the first equality, it suffices to show that
$$
\calgp(\pdg, \cA)_{O_1\cup O_2}\subseteq \calgp(\pdg, \cA)_{O_1}+\calgp(\pdg, \cA)_{O_2}.$$
Now suppose $T\in \calgp(\pdg, \cA)_{O_1\cup O_2}$. Take a continuous partition of unity
$\{\varphi_1, \varphi_2\}$ on $O_1\cup O_2$ subordinate to the open over $\{O_1, O_2\}$ of $O_1\cup O_2$.
Define two functions
$$
T_1, \; T_2: \;\; \Gamma_d\times \Gamma_d \longrightarrow \cK_p\otimes_p \cA
$$
by
$$T_1(x, y)(m)=\varphi_1(m) \Big( T(x, y)(m) \Big),$$
$$T_2(x, y)(m)=\varphi_2(m) \Big( T(x, y)(m) \Big) $$
for $x, y\in \Gamma_d$ and $m\in M$.

Then $T_1\in \calgp(\pdg, \cA)_{O_1}$, $T_2\in \calgp(\pdg, \cA)_{O_2}$, and
$$
T=T_1+T_2\in \calgp(\pdg, \cA)_{O_1}+\calgp(\pdg, \cA)_{O_2}
$$
as desired.
\end{proof}

It would be convenient to introduce the following notion associated with the coarse embedding
$f: \Gamma\to M$.

\begin{defn}\label{gammarseperate} Let $r>0$. A family of open subsets $\{O_i\}_{i\in J}$ of $M$ is said to be
{\em $(\Gamma, r)$-separate} if
\begin{enumerate}
\item $O_i\cap O_j=\emptyset $ if $i\neq j$;
\item there exists $\gamma_i\in \Gamma$ such that $O_i\subseteq B(f(\gamma_i), r)\subset M$ for each $i\in J$.
\end{enumerate}
\end{defn}

\begin{lem}\label{union-iso}
If $\{O_i\}_{i\in J}$ is a family of $(\Gamma, r)$-separate open subsets of $M$, then
$$
e_*: \lim_{d\to\infty}K_*(\clp(\pdg,\cA)_{\sqcup_{i\in J} O_i})
\to\lim_{d\to\infty}K_*(\cp(\pdg,\cA)_{\sqcup_{i\in J} O_i})
$$
is an isomorphism, where ${\sqcup_{i\in J} O_i}$ is the (disjoint) union of $\{O_i\}_{i\in J}$.
\end{lem}

We will prove Lemma \ref{union-iso} in the next section. Granting Lemma \ref{union-iso} for the moment, we are able to prove Theorem \ref{twisted-iso}.

\begin{proof}[Proof of Theorem \ref{twisted-iso}](\cite{Yu00}). For any $r>0$, we define $O_r\subset M$ by
$$ O_r=\bigcup_{\gamma\in\Gamma}B(f(\gamma),r), $$
where $f:\Gamma\to M$ is the coarse embedding and $B(f(\gamma),r)=\{p\in M:  d(p, f(\gamma))<r\}$.
\par
For any $d>0$, if $r<r'$ then $\cspace_{O_r}\subseteq \cspace_{O_{r'}}$ and $\clspace_{O_r}\subseteq \clspace_{O_{r'}}$.
By definition, we have
$$\cspace=\lim_{r\to\infty}\cspace_{O_r},$$
$$\clspace=\lim_{r\to\infty}\clspace_{O_r}.$$
\par
On the other hand, for any $r>0$, if $d<d'$ then $\Gamma_d\subseteq \Gamma_{d'}$ in
$\pdg\subseteq P_{d'}(\Gamma)$ so that we have natural inclusions
$\cspace_{O_r}\subseteq \cp(P_{d'}(\Gamma), \cA)_{O_r}$ and
$\clspace_{O_r}\subseteq \clp(P_{d'}(\Gamma), \cA)_{O_r}$.
These inclusions induce the following commuting diagram
\[
\scriptsize
\xymatrix
{
  &  K_*(\clp(P_{d'}(\Gamma), \cA)_{O_r}) \ar[rr]^{e_*} \ar'[d][dd]
      &  & K_*(\cp(P_{d'}(\Gamma), \cA)_{O_r})  \ar[dd]        \\
  K_*(\clspace_{O_r})  \ar[ur]\ar[rr]^{\mbox{\qquad\qquad $e_*$}} \ar[dd]
      &  & K_*(\cspace_{O_r}) \ar[ur]\ar[dd] \\
  &  K_*(\clp(P_{d'}(\Gamma), \cA)_{O_{r'}}) \ar'[r]^{\mbox{\qquad\qquad $e_*$}}[rr]
      &  & K_*(\cp(P_{d'}(\Gamma), \cA)_{O_{r'}})                \\
  K_*(\clspace_{O_{r'}})  \ar[rr]^{e_*}\ar[ur]
      &  & K_*(\cspace_{O_{r'}}) \ar[ur]        }
\]
\\
which allows us to change the order of limits from $\displaystyle\lim_{d\to\infty}\lim_{r\to\infty}$ to
$\displaystyle\lim_{r\to\infty}\lim_{d\to\infty}$ in the second piece of the following commuting diagram
\[
\xymatrix{
  \displaystyle\lim_{d\to\infty}K_*(\clspace)      \ar[d]_{\cong} \ar[r]^{e_*}        &
        \displaystyle\lim_{d\to\infty}K_*(\cspace)       \ar[d]_{\cong}                     \\
  \displaystyle\lim_{d\to\infty}\lim_{r\to\infty}K_*(\clspace_{O_r})      \ar[d]_{\cong}     \ar[r]^{e_*}     &
        \displaystyle\lim_{d\to\infty}\lim_{r\to\infty}K_*(\cspace_{O_r}) \ar[d]_{\cong}    \\
  \displaystyle\lim_{r\to\infty}\lim_{d\to\infty}K_*(\clspace_{O_r})      \ar[r]^{e_*}                        &
        \displaystyle\lim_{r\to\infty}\lim_{d\to\infty}K_*(\cspace_{O_r})   }
\]
So, to prove Theorem \ref{twisted-iso}, it suffices to show that, for any $r>0$,
$$
e_*:\; \lim_{d\to\infty}K_*(\clspace_{O_r})\to\lim_{d\to\infty}K_*(\cspace_{O_r})
$$
is an isomorphism.

Let $r>0$. Since $\Gamma$ has bounded geometry and $f:\Gamma\to M$ is a
coarse embedding, there exist finitely many mutually disjoint subsets of $\Gamma$,
say $\Gamma_k:=\{\gamma_i: i\in J_k\}$ with some index set $J_k$ for $k=1, 2, \cdots, k_0$,
such that $\Gamma=\bigsqcup_{k=1}^{k_0} \Gamma_k$ and, for each $k$,
$d(f(\gamma_i), f(\gamma_j))>2r$ for distinct elements $\gamma_i, \gamma_j$ in $\Gamma_k$.

For each $k=1, 2, \cdots, k_0$, let
$$
O_{r,k}=\bigcup_{i\in J_k} B(f(\gamma_i),r).
$$
Then $O_r=\bigcup^{k_0}_{k=1}O_{r,k}$ and each $O_{r,k}$, or an intersection of several $O_{r,k}$,
is the union of a family of $(\Gamma, r)$-separate (Definition \ref{gammarseperate}) open subsets of $M$.

Now Theorem \ref{twisted-iso} follows from Lemma \ref{union-iso} together with a Mayer-Vietoris sequence argument by using Lemma \ref{decomposition}.
\end{proof}

\section{Strong Lipschitz homotopy invariance}
In this section, we shall present Yu's arguments about strong Lipschitz homotopy invariance for $K$-theory of the
twisted localization algebras \cite{Yu00}, and prove Lemma \ref{union-iso} of the previous section.
\par
Let $f: \Gamma\to M$ be a coarse embedding of a bounded geometry discrete metric space $\Gamma$ into a
simply connected complete Riemannian manifold $M$ of nonpositive sectional curvature, and let $r>0$. Let
$\{O_i\}_{i\in J}$ be a  family of $(\Gamma, r)$-separate open subsets of $M$, i.e.,
\begin{enumerate}
\item $O_i\cap O_j=\emptyset $ if $i\neq j$;
\item there exists $\gamma_i\in \Gamma$ such that
$O_i\subseteq B(f(\gamma_i), r)\subset M$ for each $i\in J$.
\end{enumerate}
For $d>0$, let $X_i$, $i\in J$,  be a family of closed subsets of $\pdg$ such that $\gamma_i\in X_i$ for every $i\in J$
and $\{X_i\}_{i\in J}$ is uniformly bounded in the sense that there exists $r_0>0$ such that
$diameter(X_i)\leq r_0$ for each $i\in J$. In particular, we will consider the following three cases of
$\{X_i\}_{i\in J}$:
\begin{enumerate}
\item $X_i=B_{P_d(\Gamma)}(\gamma_i, R):=\{x\in\pdg:\; d(x,\gamma_i)\leq R\}$, for some common $R>0$ for all $i\in J$;
\item $X_i=\Delta_i$, a simplex in $\pdg$ with $\gamma_i\in \Delta_i$ for each $i\in J$;
\item $X_i=\{\gamma_i\}$ for each $i\in J$.
\end{enumerate}

For each $i\in J$, let $\cA_{O_i}$ be the subalgebra of $\cA=C_0(M, \Cliff_{\bbc}(TM))$ generated by those
functions whose supports are contained in $O_i$. We define
\begin{align*}
A(X_i:i\in J) & = \prod_{i\in J}  \; \cp(X_i)\otimes\cA_{O_i}\\
&=\left\{ \; {\bigoplus_{i\in J}} T_i   \; \left|\;\;  T_i\in \cp(X_i)\otimes\cA_{O_i}, \;
                            \sup_{i\in J} \|T_i\| <\infty \right. \right\}
\end{align*}

Similarly we define $A_L(X_i:i\in J)$ to be the subalgebra of
$$
\left\{ \; {\bigoplus_{i\in J}} \; b_i   \; \left|\;\;  b_i\in \clp(X_i)\otimes\cA_{O_i}, \; \sup_{i\in J} \|b_i\| <\infty \right. \right\}
$$
generated by elements ${\bigoplus_{i\in J}} \; b_i $ such that
\begin{enumerate}
\item the function
                $$ {\bigoplus_{i\in J}}\; b_i : \; \mathbb{R}_+ \to \prod_{i\in J}  \; \cp(X_i)\otimes\cA_{O_i}  $$
is uniformly norm-continuous in $t\in \mathbb{R}_+$.
\item there exists a bounded function $c(t)$ on $\mathbb{R}_+$ with $\lim_{t\to \infty} c(t)=0$ such that
                $(b_i(t))(x, y)=0$ whenever $d(x, y)>c(t)$ for all $i\in J$, $x, y\in X_i$ and $t\in \mathbb{R}_+$.
\end{enumerate}

For each natural number $s>0$, let $\Delta_i{(s)}$ be the simplex with vertices
$\{\gamma\in\Gamma: \; d(\gamma, \gamma_i)\leq s\}$ in $\pdg$ for $d>s$.
\par
\begin{lem}\label{decomposition}Let  $O=\sqcup_{i\in J} O_i$ be the (disjoint) union of  a family of $(\Gamma, r)$-separate
open subsets $\{O_i\}_{i\in J}$ of $M$ as above. Then
\begin{enumerate}
\item \qquad $\cspace_O
                       \;\; \cong  \;\;  \displaystyle\lim_{R\to \infty}  \;  A(\{x\in \pdg: d(x,\gamma_i)\leq R\}:i\in J)$;
\item \qquad $\clspace_O
                       \;\;  \cong \;\;  \displaystyle\lim_{R\to \infty} \; A_L(\{x\in \pdg: d(x,\gamma_i)\leq R\}:i\in J)$;
\item $\displaystyle\lim_{d\to\infty}\cspace_O
                       \;\; \cong \;\;  \lim_{s\to \infty}  \;  A(\Delta_i{(s)}:i\in J)$;
\item $\displaystyle\lim_{d\to \infty}\clspace_O
                       \;\;  \cong \;\;  \displaystyle\lim_{s\to\infty}  \;  A_L(\Delta_i{(s)}:i\in J)$.
\end{enumerate}
\end{lem}

\begin{proof}(\cite{Yu00})
Let $\cA_O$ be the subalgebra of $\cA=C_0(M,\Cliff_{\bbc}(TM))$ generated by elements whose supports
are contained in $O$. Let $\calh_O=L^2(O,\Cliff_{\bbc}(TM))$ and $\calh_{O,\infty,p}$ be the $\lp$-direct sum of infinite copies of $\calh_O$ with the $\lp$-norm
$$
\norm{(f_1,\cdots,f_n)}_p=\sqrt[p]{\sum^n_{i=1}\norm{f_i}^p},\quad\quad \text{for }f_1,\cdots,f_n\in \calh_O.
$$

$\calk_p\otimes_p\cA_O$ acts on $\calh_{O,\infty,p}$ and $\cp(\pdg,\cala)_O$ acts on $\lp(\Gamma_d, \calh_{O,\infty,p})$. We have a decomposition
$$
\lp(\Gamma_d, \calh_{O,\infty,p})=\bigoplus_{i\in J} \lp(\Gamma_d, \calh_{O_i,\infty,p}).
$$
Each $T\in \calgp(\pdg,\cA)_O$ has a corresponding decomposition
$$
T=\bigoplus_{i\in J} \;  T_i
$$
such that there exists $R>0$ for which each $T_i$ is supported on
$$
\{(x,y,p): \; p\in O_i,\; x,y\in\Gamma_d, \; d(x,\gamma_i)\leq R, \; d(y,\gamma_i)\leq R\}.
$$
On the other hand, the Banach algebra
$\cp(\{x\in\pdg:d(x,\gamma_i)\leq R\})\otimes_p \cA_{O_i}$ acts on
$$
\ell^p(\{x\in\Gamma_d:d(x,\gamma_i)\leq R\},\calh_{O_i,\infty,p}),
$$
so that on $\ell^p(\Gamma_d,\calh_{O_i,\infty,p})$, for each $R>0$,
the algebra $A(\{x\in\pdg:d(x,\gamma_i)\leq R\}:i\in J)$ can be
represented as a subalgebra of $\cspace_O$. In this way,  the decomposition
$T=\oplus_{i\in J}T_i$ induces a Banach algebra isomorphism
$$
\cspace_O\cong\displaystyle\lim_{R\to\infty}A(\{x\in\pdg:d(x,\gamma_i)\leq R \}: i\in J)
$$
as desired in (1). Then (2),(3),(4) follows easily from (1).
\end{proof}

Now we turn to recall the notion of strong Lipschitz homotopy \cite{Yu97, Yu98, Yu00}.

Let $\{Y_i\}_{i\in J}$ and $\{X_i\}_{i\in J}$ be two families of uniformly bounded closed subspaces of
$\pdg$ for some $d>0$ with $\gamma_i\in X_i$, $\gamma_i\in Y_i$ for every $i\in J$.
A map $g: \; \bigsqcup_{i\in J}X_i\to \bigsqcup_{i\in J}Y_i$ is said to be $Lipschitz$ if
\begin{enumerate}
\item $g(X_i)\subseteq Y_i$ for each $i\in J$;
\item there exists a constant $c$, independent of $i\in J$, such that
$$ d(g(x),g(y))\leq c \; d(x,y) $$
for all $x,y\in X_i$, $i\in J$.
\end{enumerate}

Let $g_1,g_2$ be two Lipschitz maps from $\bigsqcup_{i\in J}X_i$ to $\bigsqcup_{i\in J}Y_i$.
We say $g_1$ is {\it strongly Lipschitz homotopy} equivalent to $g_2$ if there exists a continuous map
$$
F: \; [0,1]\times (\sqcup_{i\in J}X_i)\to \sqcup_{i\in J}Y_i
$$
such that
\begin{enumerate}
\item $F(0,x)=g_1(x)$, $F(1,x)=g_2(x)$ for all $x\in\sqcup_{i\in J}X_i$;
\item there exists a constant $c$ for which $ d(F(t,x),F(t,y))\leq c \; d(x,y)$
for all $x,y\in X_i$, $t\in [0,1]$, where $i$ is any element in $J$;
\item $F$ is equicontinuous in $t$, i.e., for any $\varepsilon>0$ there exists $\delta>0$ such that
$d(F(t_1,x),F(t_2,x))<\varepsilon $ for all $x\in\sqcup_{i\in J}X_i$ if $|t_1-t_2|<\delta$.
\end{enumerate}

We say $\{X_i\}_{i\in J}$ is {\it strongly Lipschitz homotopy} equivalent to $\{Y_i\}_{i\in J}$ if there
exist Lipschitz maps $g_1:\sqcup_{i\in J}X_i\to \sqcup_{i\in J}Y_i$ and
$g_2:\sqcup_{i\in J}Y_i\to \sqcup_{i\in J}X_i$ such that $g_1g_2$ and $g_2g_1$ are
respectively strongly Lipschitz homotopy equivalent to identity maps.

Define $A_{L,0}(X_i:i\in J)$ to be the subalgebra of $A_L(X_i:i\in J)$ consisting of elements
$\oplus_{i\in J}b_i(t)$ satisfying $b_i(0)=0$ for all $i\in J$.

\begin{lem}[\cite{Yu00}]\label{lipschitz-iso}
If $\{X_i\}_{i\in J}$ is strongly Lipschitz homotopy equivalent to $\{Y_i\}_{i\in J}$
then $K_*(A_{L,0}(X_i:i\in J))$ is isomorphic to $K_*(A_{L,0}(Y_i:i\in J))$.
\end{lem}

Let $e$ be the evaluation homomorphism from $A_L(X_i:i\in J)$  to $A(X_i:i\in J)$ given by
$\oplus_{i\in J} \; g_i(t) \mapsto \oplus_{i\in J}g_i(0)$.

\begin{lem}[\cite{Yu00}]\label{family-iso}
Let $\{\gamma_i\}_{i\in J}$ be as above, i.e.,
$O_i\subseteq B(f(\gamma_i), r)\subset M$ for each $i$. If $\{\Delta_i\}_{i\in
J}$ is a family of simplices in $\pdg$ for some $d>0$ such that
$\gamma_i\in \Delta_i$ for all $i\in J$, then
$$ e_*:K_*(A_L(\Delta_i:i\in J))\to K_*(A(\Delta_i:i\in J)) $$
is an isomorphism.
\end{lem}
\begin{proof} (\cite{Yu00}){\bf .}
Note that $\{\Delta_i\}_{i\in J}$ is strongly Lipschitz homotopy equivalent to
$\{\gamma_i\}_{i\in J}$. By an argument of Eilenberg swindle,  we have
$K_*(A_{L,0}(\{\gamma_i\}:i\in J))=0$. Consequently, Lemma \ref{family-iso} follows from Lemma \ref{lipschitz-iso} and the six term exact sequence of Banach algebra $K$-theory.
\end{proof}

We are now ready to give a proof to Lemma \ref{union-iso} of the previous section.

\begin{proof}[Proof of Lemma \ref{union-iso}, \cite{Yu00}]
By Lemma \ref{decomposition} we have the following commuting diagram
\[
\xymatrix{
  \displaystyle\lim_{d\to\infty}\clspace_{\sqcup_{i\in J} O_i}           \ar[d]_{\cong}          \ar[r]^{e}
            &    \displaystyle\lim_{d\to\infty}\cspace_{\sqcup_{i\in J} O_i}          \ar[d]^{\cong}                   \\
  \displaystyle\lim_{s\to\infty}A_L(\Delta_i{(s)}_i: \; i\in J)           \ar[r]^{e}
            &    \displaystyle\lim_{s\to\infty}A(\Delta_i{(s)}_i: \; i\in J)
     }
\]
which induces the following commuting diagram at $K$-theory level
\[
\xymatrix{
  \displaystyle\lim_{d\to\infty} K_*\Big(\clspace_{\sqcup_{i\in J} O_i}\Big)           \ar[d]_{\cong}          \ar[r]^{e_*}
            &    \displaystyle\lim_{d\to\infty} K_*\Big(\cspace_{\sqcup_{i\in J} O_i}\Big)          \ar[d]^{\cong}                   \\
  \displaystyle\lim_{s\to\infty} K_*\Big(A_L(\Delta_i{(s)}: \; i\in J)\Big)           \ar[r]^{e_*}
            &    \displaystyle\lim_{s\to\infty} K_*\Big(A(\Delta_i{(s)}: \; i\in J)\Big) .
     }
\]
Now Lemma \ref{union-iso} follows from Lemma \ref{family-iso}.
\end{proof}

\section{Almost flat Bott elements and Bott  maps}
In this section, we shall construct uniformly almost flat Bott generators for a simply connected complete Riemannian
manifold with nonpositive sectional curvature, and define a Bott map from the $K$-theory of the Roe
algebra to the $K$-theory of the twisted Roe algebra and another Bott map between the
$K$-theory of corresponding localization algebras.
We show that the Bott map from the $K$-theory of the $\lp$-localization algebra to the $K$-theory of the twisted
$\lp$-localization algebra is an isomorphism (Theorem \ref{twisted-bott}).

Let $M$ be a simply connected complete Riemannian manifold with nonpositive sectional curvature.
As remarked at the beginning of Section 4, without loss of generality, we assume in the following
$\dim (M)=2n$ for some integer $n>0$.

Recall that $\cA:=C_0(M,\Cliff_{\bbc}(TM))$ and $\dim M=2n$,  the exponential map
$$\exp_x: \; T_x M \cong \mathbb{R}^{2n} \to M $$
at any point $x\in M$ induces an isomorphism
$$ C_0(M,\cliff_{\bbc}(TM))\cong C_0(\mathbb{R}^{2n})\otimes \cM_{2^n}(\mathbb{C}). $$
Similarly, we define $\cB:=C_b(M, \cliff_{\bbc}(TM))$ to be the Banach algebra of all bounded functions $a$ on $M$ with
$a(x)\in \Cliff(T_x M)$ at all $x\in M$.

Let $x\in M$. For any $z\in M$, let $\sigma: [0, 1]\to M$ be the unique geodesic such that
$$\sigma(0)=x,\qquad\sigma(1)=z.$$
Let $v_x(z):=\frac{\sigma'(1)}{\|{\sigma'(1)}\|} \in T_z M$.  For any $c>0$, take a continuous function
$\phi_{x,c}:M\to [0,1]$ satisfying
\begin{equation}
\phi_{x,c}(z)=\left\{
\begin{array}{ccc}
0, & \mbox{if}\ d(x,z)\leq \frac{c}{2};  \\
1, & \mbox{if}\ d(x,z)\geq c.
\end{array}\right.
\end{equation}
For any $z\in M$, let
$$ f_{x,c}(z):=\phi_{x,c}(z)\cdot v_x(z)\in T_zM. $$
Then $f_{x,c}\in \calb$. The following result describes certain ``uniform almost flatness" of
the functions $f_{x, c}$ ($x\in M$, $c>0$).

\begin{lem}\label{flat}For any $R>0$ and $\varepsilon>0$, there exist a constant $c>0$ and a family of continuous
function $\{\phi_{x,c}\}_{x\in M}$  satisfying the above condition (1) such that, if $d(x,y)<R$, then
$$\sup_{z\in M}\|{f_{x,c}(z)-f_{y,c}(z)}\|_{T_zM}< \varepsilon. $$
\end{lem}
\begin{proof}Let $c=\frac{2R}{\varepsilon}$. For any $x\in M$, define $\phi_{x,c}:M\to [0,1]$ by
\[
\phi_{x,c}(z)=\left\{
\begin{array}{ll}
0, & \mbox{ if }  d(x,z)\leq \frac{R}{\varepsilon};        \\
\frac{\varepsilon}{R}d(x, z)-1,   &  \mbox{ if }  \frac{R}{\varepsilon}\leq d(x,z)\leq \frac{2R}{\varepsilon};  \\
1, & \mbox{ if }  d(x,z)\geq \frac{2R}{\varepsilon}.
\end{array}\right.
\]
Let $x,y\in M$ such that $d(x,y)<R$. Then we have several cases for the position of $z\in M$ with respect to
$x, y$.
\par
Consider the case where $d(x,z)>c=\frac{2R}{\varepsilon}$ and $d(y,z)>c=\frac{2R}{\varepsilon}$.
Since $\phi_{x,c}(z)=\phi_{y,c}(z)=1$, we have
$$f_{x,c}(z)-f_{y,c}(z)=v_x(z)-v_y(z).$$
Without loss of generality, assume $d(x, z)\leq d(y, z)$. Then there exists a unique point $y'$ on the unique
geodesic connecting $y$ and $z$ such that $d(y', z)=d(x, z)$. Then $d(y', y)<R$ since $d(x, y)<R$, so that
$d(x, y')<2R$.
\par
Let $\exp^{-1}_z:M\to T_zM$ denote the inverse of the exponential map
$$\exp_z: T_zM\to M $$
at $z\in M$. Then we have
\\ \indent ($\alpha$) $\|\exp^{-1}_z(x)\|=d(x,z)=d(y',z)=\|\exp^{-1}_z(y')\|>c=\frac{2R}{\varepsilon}$;
\\ \indent ($\beta$)  $\|\exp^{-1}_z(x)-\exp^{-1}_z(y')\|\leq d(x,y')<2R$, since $M$ has nonpositive sectional curvature;
\\ \indent ($\gamma$) $v_x(z)=-\frac{\exp^{-1}_z(x)}{\|{\exp^{-1}_z(x)}\|}$
                    and $v_y(z)=-\frac{\exp^{-1}_z(y')}{\|{\exp^{-1}_z(y')}\|}$.
\\
Hence, for any $z\in M$, we have
$$\|{f_{x,c}(z)-f_{y,c}(z)}\| = \|{v_x(z)-v_y(z)}\| <2R/(2R/\varepsilon)=\varepsilon  $$
whenever $d(x, y)<R$. Similarly, we can check the inequality in other cases where $z\in M$ satisfies either
$d(x,z)\leq c$ or $d(y,z)\leq c$.
\end{proof}

Now let's consider the short exact sequence
$$
0\longrightarrow \cA\longrightarrow\cBp \stackrel{\pi}{\longrightarrow} \cBp/\cA \longrightarrow0,
$$
where $\cA=C_0(M,\cliff_{\bbc}(TM))$ and $\cBp=C_b(M,\cliff_{\bbc}(TM))$.
For any $f_{x, c}$ ($x\in M$, $c>0$) constructed above, it is easy to see that
$[f_{x,c}]:=\pi(f_{x,c})$ is invertible in $\cBp/\cA$ with its inverse $[-f_{x,c}]$. Thus
$[f_{x,c}]$ defines an element in $K_1(\cBp/\cA)$. With the help of the index map
$$
\partial: K_1(\cBp/\cA)\to K_0(\cA),
$$
we obtain an element $\partial([f_{x,c}])$ in
$$
K_0(\cA)=K_0\Big( C_0(M,\cliff_{\bbc}(TM)) \Big) \cong K_0 \Big( C_0(\mathbb{R}^{2n})\otimes \cM_{2^n}(\mathbb{C}) \Big)
        \cong\mathbb{Z}.
        $$
It follows from the construction of $f_{x,c}$ that, for every $x\in M$ and $c>0$, $\partial([f_{x,c}])$ is just the Bott generator of $K_0(\cA)$.

The element $\partial([f_{x,c}])$ can be expressed explicitly as follows. Let
\begin{align*}
W_{x,c}&=\matr{1}{f_{x,c}}{0}{1}\matr{1}{0}{f_{x,c}}{1}\matr{1}{f_{x,c}}{0}{1}\matr{0}{-1}{1}{0} \; ,\\
b_{x,c}&=W_{x,c}\matr{1}{0}{0}{0}W_{x,c}^{-1} \; ,\\
b_0&=\matr{1}{0}{0}{0}\; .
\end{align*}

Then both $b_{x,c}$ and $b_0$ are idempotents in $\cM_2(\cA^+)$, where $\cA^+$ is the algebra jointing a unit
to $\cA$.  It is easy to check that
$$
b_{x,c}-b_0\in C_c(M,\cliff_{\bbc}(TM))\otimes \cM_2(\mathbb{C}),
$$
the algebra of $2\times 2$ matrices  of compactly supported continuous functions, with
$$ \supp(b_{x,c}-b_0)\subset B_M(x,c):=\{z\in M:d(x,z)\leq c\}, $$
where for a matrix $a=\matr{a_{11}}{a_{12}}{a_{21}}{a_{22}}$ of functions on $M$ we define the support of $a$ by
$$\supp (a)= \bigcup^2_{i, j=1} \supp(a_{i, j}).$$
Now we have the explicit expression
$$ \partial([f_{x,c}])=[b_{x,c}]-[b_0]\in K_0(\cA). $$

\begin{lem}[Uniform almost flatness of the Bott generators]\label{bottflat}
The family of idempotents $\{b_{x,c}\}_{x\in M,c>0}$ in $\cM_2(\cA^+)=C_0(M,\cliff_{\bbc}(TM))^+\otimes \cM_2(\mathbb{C})$
constructed above are uniformly almost flat in the following sense:
\\ \indent for any $R>0$ and $\varepsilon>0$, there exist $c>0$ and a family of continuous functions
$\Big\{ \phi_{x,c}: \; M\to [0,1] \Big\}_{x\in M}$ such that, whenever $d(x,y)<R$, we have
$$ \sup_{z\in M} \|{b_{x,c}(z)-b_{y,c}(z)}\|_{\cliff_\bbc(T_z M)\otimes \cM_2(\mathbb{C})}<\varepsilon, $$
where $b_{x, c}$ is defined via $W_{x, c}$ and $f_{x, c}=\phi_{x, c} v_x$ as above, and
$\cliff_\bbc(T_z M)$ is the complexified Clifford algebra of the tangent space $T_z M$.
\end{lem}
\begin{proof} Straightforward from Lemma \ref{flat}.    \end{proof}

It would be convenient to introduce the following notion:
\par
\begin{defn} For $R>0, \varepsilon>0, c>0$, a family of idempotents $\{b_x\}_{x\in M}$ in
$\cM_2(\cA^+)=C_0(M,\cliff_{\bbc}(TM))^+\otimes \cM_2(\mathbb{C})$ is said to be $(R, \varepsilon; c)$-flat if
\begin{enumerate}
\item for any  $x, y\in M$ with $d(x, y)<R$ we have
      $$
      \sup_{z\in M} \|{b_{x}(z)-b_{y}(z)}\|_{\cliff_\bbc(T_z M)\otimes \cM_2(\mathbb{C})}<\varepsilon.
      $$
\item $b_{x}-b_0\in C_c(M,\cliff_{\bbc}(TM))\otimes \cM_2(\mathbb{C})$ and
      $$
      \supp(b_{x}-b_0)\subset B_M(x,c):=\{z\in M:d(x,z)\leq c\}.
      $$
\end{enumerate}
\end{defn}

{\bf Construction of the Bott map $\beta_*$:}

Now we shall use the above almost flat Bott generators for
$$
K_0(\cA)=K_0\Big(C_0(M,\cliff_{\bbc}(TM))\Big)
$$
 to construct a ``Bott map"
$$
\beta_*: \; K_*(\cp(\pdg))\to K_*(\cspace).
$$

To begin with, we give a representation of $\cp(\pdg)$ on $\lp(\Gamma_d,\lp)$, where $\Gamma_d$ is the
countable dense subset of $\pdg$ and $H_0$ is the Hilbert space as in the definition of $\cspace$.

Let $\calgp(\pdg)$ be the algebra of functions
$$
Q: \; \Gamma_d\times \Gamma_d \to \cK_p
$$
such that
\\ \indent (1) there exists $C>0$ such that $\|Q(x,y)\|\leq C$ for all $x,y\in\Gamma_d$;
\\ \indent (2) there exists $R>0$ such that $Q(x,y)=0$ whenever $d(x,y)>R$;
\\ \indent (3) there exists $L>0$ such that for every $z\in P_d(\Gamma)$, the number of elements
in the following set
$$\{ (x, y)\in\Gamma_d\times \Gamma_d: \; d(x, z)\leq 3R, \; d(y, z)\leq 3R, \; Q(x,y)\neq 0 \}$$
is less than $L$.
\par
The product structure on $\calgp(\pdg)$ is defined by
$$
(Q_1Q_2)(x, y)=\sum_{z\in \Gamma_d} Q_1(x, z) Q_2(z, y).
$$
The algebra $\calgp(\pdg)$ acts on $\lp(\Gamma_d,\lp)$. The operator norm completion
of $\calgp(\pdg)$ with respect to this action is isomorphic to $\cp(\pdg)$ when $\Gamma$ has bounded geometry.

Note that $\cp(\pdg)$ is stable in the sense that $\cp(\pdg)\cong \cp(\pdg)\otimes \cM_k(\mathbb{C})$ for all natural
number $k$. Any element in $K_0(\cp(\pdg))$ can be expressed as the difference of the $K_0$-classes of two idempotents
in $\cp(\pdg)$. To define the Bott map $\beta_*: \; K_0(\cp(\pdg))\to K_0(\cspace)$, we need to specify the value
$\beta_*([P])$ in $K_0(\cspace)$ for any idempotent $P\in \cp(\pdg)$.

Now let $P\in \cp(\pdg)\subseteq \cB(\lp(\Gamma_d,\lp))$ be an idempotent. Denote $\norm{P}=N$. For any $0<\varepsilon_1<1/100$,
take an element $Q\in \calgp(\pdg)$ such that
$$
\|P-Q\|< \frac{\varepsilon_1}{2N+2}.
$$
Then $\|Q\|<\|P-Q\|+\|P\|<N+1$, hence
$$
\|Q-Q^2\|\leq\|Q-P\|+\|P\|\|P-Q\|+\|P-Q\|\|Q\|<\varepsilon_1
$$
and there is $R_{\varepsilon_1}>0$ such that $Q(x, y)=0$
whenever $d(x, y)>R_{\varepsilon_1}$.
For any $\varepsilon_2>0$, take by Lemma \ref{bottflat} a family of $(R_{\varepsilon_1}, \varepsilon_2; c)$-flat idempotents
$\{b_x\}_{x\in M}$ in $\cM_2(\cA^+)$ for some $c>0$. Define
$$
\widetilde{Q}, \; \widetilde{Q}_0: \; \Gamma_d\times\Gamma_d \to \cK_p\otimes_p\cA^+  \otimes \cM_2(\mathbb{C})
$$
by
$$
\widetilde{Q}(x,y)=Q(x,y)\otimes b_x
$$
and
$$
\widetilde{Q}_0(x,y)=Q(x,y)\otimes b_0,
$$
respectively, for all $(x, y)\in \Gamma_d\times \Gamma_d$, where $b_0=\matr{1}{0}{0}{0}$. Then
$$
\widetilde{Q}, \widetilde{Q}_0 \in \calgp(\pdg, \cA^+\otimes \cM_2(\mathbb{C}))
            \cong \calgp(\pdg,\cA^+)\otimes \cM_2(\mathbb{C})
$$
and
$$
\widetilde{Q}-\widetilde{Q}_0 \in \calgp(\pdg,\cA)\otimes \cM_2(\mathbb{C}).
$$

Since $\Gamma$ has bounded geometry, by the almost flatness of the Bott generators (Lemma \ref{bottflat}), we can choose $\varepsilon_1$ and $\varepsilon_2$
small enough to obtain $\widetilde{Q}, \widetilde{Q}_0$ as constructed above such that $\|\widetilde{Q}^2-\widetilde{Q}\|<1/5$ and $\|\widetilde{Q}_0^2-\widetilde{Q}_0\|<1/5$.

It follows that the spectrum of either $\widetilde{Q}$ or $\widetilde{Q}_0$ is contained in  disjoint
neighborhoods $S_0$ of $0$ and $S_1$ of $1$ in the complex plane. Let $f: S_0\sqcup S_1\to \mathbb{C}$ be a holomorphic
function such that $f(S_0)=\{0\}, f(S_1)=\{1\}$. Let $\Theta=f(\widetilde{Q})$ and
$\Theta_0=f(\widetilde{Q}_0)$. Then $\Theta$ and $\Theta_0$ are idempotents in $\cp(\pdg,\cA^+)\otimes \cM_2(\mathbb{C})$  with
$$
\Theta-\Theta_0\in \cspace\otimes \cM_2(\mathbb{C}).
$$
Note that $\cspace\otimes \cM_2(\mathbb{C})$ is a closed two-sided ideal of $\cp(\pdg,\cA^+)\otimes \cM_2(\mathbb{C})$.

At this point we need to recall the {\em difference construction} in $K$-theory of Banach algebras
introduced by Kasparov-Yu \cite{KY}. Let $J$ be a closed two-sided ideal of a Banach algebra $B$.
Let $p, q\in B^+$ be idempotents such that $p-q\in J$.
Then a difference element $D(p,q)\in K_0(J)$ associated to the pair $p,q$ is defined as follows. Let
\[
Z(p, q)=\left(
\begin{array}{cccc}
q    &  0   &  1-q  &    0    \\
1-q  &  0   &   0   &    q    \\
0    &  0   &   q   &   1-q   \\
0    &  1   &   0   &    0
\end{array}
\right)
\in \cM_4(B^+).
\]
We have
\[
(Z(p, q))^{-1}=\left(
\begin{array}{cccc}
q    &  1-q &   0   &    0    \\
0    &  0   &   0   &    1    \\
1-q  &  0   &   q   &    0    \\
0    &  q   &  1-q  &    0
\end{array}
\right)
\in \cM_4(B^+).
\]
Define
\[
D_0(p,q)=(Z(p, q))^{-1}\left(
\begin{array}{cccc}
p  &  0  &  0  &  0 \\
0  & 1-q &  0  &  0 \\
0  &  0  &  0  &  0 \\
0  &  0  &  0  &  0
\end{array}
\right)
Z(p, q).
\]
Let
\[
p_1=\left(
\begin{array}{cccc}
1  &  0  &  0  &  0 \\
0  &  0  &  0  &  0 \\
0  &  0  &  0  &  0 \\
0  &  0  &  0  &  0
\end{array}
\right).
\]
Then $D_0(p, q)\in \cM_4(J^+)$ and
$D_0(p, q)=p_1 $ modulo $\cM_4(J)$. We define the difference element
$$
D(p, q):=[D_0(p, q)]-[p_1]
$$
in $K_0(J)$.

Finally, for any idempotent $P\in \cp(\pdg)$  representing an element  $[P]$ in $K_0(\cp(\pdg))$, we define
$$
\beta_*([P])=D(\Theta, \Theta_0)\in K_0(\cspace).
$$
The correspondence $[P]\to \beta_*([P])$ extends to a homomorphism, the Bott map
$$
\beta_*: \; K_0(\cp(\pdg))\to K_0(\cspace).
$$
By using suspension, we similarly define the Bott map
$$\beta_*: \; K_1(\cp(\pdg))\to K_1(\cspace).$$

{\bf  Construction of the Bott map $(\beta_L)_*$ :}

Next we shall construct a Bott map for $K$-theory of $\lp$-localization algebras:
$$
(\beta_L)_*: \; K_*(\clp(\pdg))\to K_*(\clspace).
$$

Let $\clalgp(\pdg)$ be the algebra of all bounded, uniformly continuous functions
$$
g: \; \mathbb{R}_+ \to \calgp(\pdg) \subset \cB(\lp(\Gamma_d,\lp))
$$
with the following properties:
\begin{enumerate}
\item there exists a bounded function $R: \mathbb{R}_+\to \mathbb{R}_+$ with
                $\displaystyle\lim_{t\to \infty} R(t)=0$ such that $g(t)(x,y)=0$ whenever $d(x,y)>R(t)$ for every $t$;
\item there exists $L>0$ such that for every $z\in P_d(\Gamma)$, the number of elements
                    in the following set
                    $$\{ (x, y)\in\Gamma_d\times \Gamma_d: \; d(x, z)\leq 3R, \; d(y, z)\leq 3R, \; g(t)(x,y)\neq 0 \}$$
                    is less than $L$ for every $t\in \mathbb{R}_+$.
\end{enumerate}

The $\lp$-localization algebra $\clp(\pdg)$ is isomorphic to the norm completion of $\clalgp(\pdg)$ under the norm
$$
\|g\|_\infty:=\sup_{t\in \mathbb{R}_+} \|g(t)\|
$$
when $\Gamma$ has bounded geometry.  Note that $\clp(\pdg)$ is stable in the sense that
$\clp(\pdg)\cong \clp(\pdg)\otimes \cM_k(\mathbb{C})$ for all natural number $k$. Hence, any element in
$K_0(\clp(\pdg))$ can be expressed as the difference of the $K_0$-classes of two idempotents in $\clp(\pdg)$.
To define the Bott map $(\beta_L)_*: \; K_0(\clp(\pdg))\to K_0(\clspace)$, we need to specify the value
$(\beta_L)_*([g])$ in $K_0(\clspace)$ for any idempotent $g\in \clp(\pdg)$ representing an element
$[g]\in K_0(\clp(\pdg))$.

Now let $g\in \clp(\pdg)$ be an idempotent with $\norm{g}=N$. For any $0<\varepsilon_1<1/100$,
take an element $h\in \clalgp(\pdg)$ such that
$$
\|g-h\|_\infty< \frac{\varepsilon_1}{2N+2}.
$$
Then $\norm{h-h^2}_\infty<\varepsilon_1$ and there is a bounded function $R_{\varepsilon_1}(t)>0$
with $\displaystyle\lim_{t\to \infty} R_{\varepsilon_1}(t)=0$ such that
$h(t)(x, y)=0$ whenever $d(x, y)>R_{\varepsilon_1}(t)$ for every $t$. Let $\widetilde{R}_{\varepsilon_1}=\sup_{t\in \mathbb{R}_+} R(t)$.
For any $\varepsilon_2>0$, take by Lemma \ref{bottflat} a family of $(\widetilde{R}_{\varepsilon_1}, \varepsilon_2; c)$-flat idempotents
$\{b_x\}_{x\in M}$ in $\cM_2(\cA^+)$ for some $c>0$. Define
$$
\widetilde{h},\; \widetilde{h}_0: \; \mathbb{R}_+\to \calgp(\pdg,\cA^+)\otimes \cM_2(\mathbb{C})
$$
by
$$
\Big( \widetilde{h}(t) \Big) (x, y) = \Big( h(t)(x,y) \Big) \otimes b_x
                        \in \cA^+\otimes \cK_p \otimes \cM_2(\mathbb{C}),
$$
$$
\Big( \widetilde{h}_0(t) \Big) (x, y) = \Big( h(t)(x,y) \Big) \otimes \scriptsize{\matr{1}{0}{0}{0}}
                        \in \cA^+\otimes \cK_p \otimes \cM_2(\mathbb{C})
$$
for each $t\in \mathbb{R}_+$. Then we have
$$
\widetilde{h}, \; \widetilde{h}_0 \; \in \;  \clalgp(\pdg,\cA^+)\otimes \cM_2(\mathbb{C})
$$
and
$$
\widetilde{h}-    \widetilde{h}_0 \; \in  \; \clalgp(\pdg,\cA)\otimes \cM_2(\mathbb{C}).
$$
Since $\Gamma$ has bounded geometry, by the almost flatness of the Bott generators,
we can choose $\varepsilon_1$ and $\varepsilon_2$
small enough to obtain $\widetilde{h}, \widetilde{h}_0$, as constructed above, such that
$\|\widetilde{h}^2-\widetilde{h}\|_\infty<1/5$ and $\|\widetilde{h}_0^2-\widetilde{h}_0\|<1/5$.
The spectrum of either $\widetilde{h}$ or $\widetilde{h}_0$ is contained in disjoint neighborhoods
$S_0$ of $0$ and $S_1$ of $1$ in the complex plane.
Let $f: S_0\sqcup S_1\to \mathbb{C}$ be the
function such that $f(S_0)=\{0\}, f(S_1)=\{1\}$. Let $\eta=f(\widetilde{h})$ and
$\eta_0=f(\widetilde{h}_0)$. Then $\eta$ and $\eta_0$ are idempotents in
$\clp(\pdg,\cA^+)\otimes \cM_2(\mathbb{C})$  with
$$
\eta-\eta_0\in \clspace\otimes \cM_2(\mathbb{C}).
$$
Thanks to the difference construction, we define
$$
(\beta_L)_*([g])=D(\eta, \eta_0)\in K_0(\clspace).
$$
This correspondence $[g]\mapsto (\beta_L)_*([g])$ extends to a homomorphism, the Bott map
$$
(\beta_L)_*: \; K_0(\clp(\pdg))\to K_0(\clspace).
$$
By suspension, we similarly define
$$
(\beta_L)_*: \; K_1(\clp(\pdg))\to K_1(\clspace).
$$
This completes the construction of the Bott map $(\beta_L)_*$.

It follows from the constructions of $\beta_*$ and $(\beta_L)_*$, we
have the following commuting diagram
\[
\xymatrix{
  K_*(\clp(\pdg)) \ar[d]_{e_*} \ar[r]^{(\beta_L)_*}
                &  K_*(\clspace)   \ar[d]^{e_*} \\
  K_*(\cp(\pdg))   \ar[r]^{\beta_*} & K_*(\cspace)
     }
\]

\begin{thm}\label{twisted-bott} For any $d>0$, the Bott map
$$
(\beta_L)_*: \; K_*(\clp(\pdg))\to K_*(\clspace)
$$
is an isomorphism.
\end{thm}
\begin{proof} Note that $\Gamma$ has bounded geometry, and both the $\lp$-localization algebra and
the twisted $\lp$-localization algebra have strong Lipschitz homotopy invariance at the $K$-theory level.
By a Mayer-Vietoris sequence argument and induction on the dimension of the skeletons \cite{Yu97, CW02}, the general
case can be reduced to the $0$-dimensional case, i.e., if $D\subset \pdg$ is a
$\delta$-separated subspace (meaning $d(x,y)\geq \delta$ if $x\neq y\in D$) for some $\delta>0$, then
$$(\beta_L)_*: \; K_*(\clp(D))\to K_*(\clp(D,\cA))$$
is an isomorphism. But this follows from the facts that
$$K_*(\clp(D))\cong \prod_{\gamma\in D} K_*(\clp(\{\gamma\})),$$
$$K_*(\clp(D,\cA))\cong\prod_{\gamma\in D} K_*(\clp(\{\gamma\}, \cA))$$
and that $(\beta_L)_*$ restricts to an isomorphism from $K_*(\clp(\{\gamma\}))\cong K_*(\cK_p)$ to
$$K_*(\clp(\{\gamma\}, \cA)) \cong K_*(\cK_p\otimes \cA)$$
at each $\gamma\in D$ by the classic Bott periodicity.
\end{proof}

\section{Proof of the Main Theorem}
\begin{proof}[Proof of Theorem \ref{main}] We have the commuting diagram
\[
\xymatrixcolsep{3pc}\xymatrix{
            \displaystyle\lim_{d\to\infty}K_*(\clp(P_d(\Gamma))) \ar[d]_{e_*}\ar[r]^{(\beta_L)_*}_{\cong}
            &   \displaystyle\lim_{d\to\infty}K_*(\clspace)  \ar[d]^{e_*}_{\cong}                     \\
\displaystyle\lim_{d\to\infty}K_*(\cp(P_d(\Gamma)))\ar[r]^{\beta_*}
   &       \displaystyle\lim_{d\to\infty}K_*(\cspace).
  }
\]
Hence, $\beta_*\circ e_* =e_*\circ(\beta_L)_*$ It follows from Theorem \ref{twisted-iso}
and Theorem \ref{twisted-bott} that $\beta_*\circ ind$ is an isomorphism. Consequently, the index map
$$
e_*: \; \displaystyle\lim_{d\to\infty}K_*(\clp(P_d(\Gamma))) \to
            \displaystyle\lim_{d\to\infty}K_*(\cp(P_d(\Gamma)))\cong K_*(\cp(\Gamma))$$
is injective.
\end{proof}

redpajama_set_name: RedPajamaArXiv


Green Deer is Red Deer's city-wide spring cleanup program. Residents, groups and businesses take on the task of picking up litter and ensuring Red Deer is clean and green.
There are many ways to get involved in your neighbourhood and make connections that contribute to residents overall quality of life.
Is there a green space in Red Deer that's special to you? Why not adopt it?
There are many events happening in Red Deer that make it a great place to live and visit.


text:
Virginia Rucker Walter Poetry Prize for an Undergraduate Student Poet
The Virginia Rucker Walter Poetry Prize for an Undergraduate Poet is named for Virginia Rucker Walter, who was a poet and student at UGA in the 1980s. Virginia Walter was killed by a drunk driver before she was able to graduate, but her family presents this prize in honor of her memory and in celebration of her love of writing poetry.
Winner: Karla Nemanic, for the poem, "Me Rajaron de las Raíces"
From Lindsay Tigue, judge: "Me Rajaron de las Raíces" simultaneously sets up the ties and differences between the speaker and the speaker's mother, who immigrated from Chile when she was nine. The poet beautifully uses the act of the mother braiding their child's hair to impart both the space and closeness between their lived experiences. The mother has "bird hands" while the speaker describes their own "chubby fingers"; they have differing appreciation for poetry. And yet they share so much. The wonderfully concrete setting of the peeling peach paint of the bathroom exists alongside the places the speaker's mother was "torn up from," even the speaker's own hair, as it's being pulled into braids becomes the Andes mountains. Just as the mountains and hair-strewn tile of the bathroom are "criss-crossed and scratched" so are mother and child connected by the history and culture they share.
Diann Blakely Poetry Prize for a Graduate Student Poet
The Diann Blakely Poetry Prize for a Graduate Student Poet is presented in honor of American poet, essayist, editor, and critic Diann Blakely.  Before passing away in 2014, Diann was known for her commitment to southern poetry and culture. She taught at Belmont University, Harvard University, Vanderbilt University, and was a former poetry editor at the Antioch Review and at New World Writing.  This award is made possible as part of the Diann Blakely Visiting Poet Fund.
Winner: Paul Cunningham, for the poem, "Factory Appetite"
From Lindsay Tigue, judge:"Factory Appetite" considers the deep history of a place—a part of Pennsylvania where the Anchor Hocking/Phoenix Glass plant is located. The poem includes references to the layered history of naming, of disaster, of environmental impact, and what it is like to be another product of a town that's become indistinguishable from its industry. The speaker "grew up . . . hungry for excess fabric," a self-indictment inextricable from capitalistic influences at work in this immediate environment and society at large. I am impressed by the way the poet zooms into the precise language of one place—"a tract of land called Appetite," while accounting for the unknown impacts of disaster writ large, wondering "how long it takes how long it takes a disaster's ashes / to sky a town with new birds." I won't soon forget this turn of phrase—"to sky a town" and to the cyclic evolution of place it represents, showing us that even as a place changes it also—like the speaker—cannot forestall future disaster.
Lindsay Tigue is the author of System of Ghosts, which was the winner of the Iowa Poetry Prize. Tigue writes poetry and prose and her work appears in Prairie Schooner, Blackbird, Verse Daily, and Hayden's Ferry Review, among other journals. She was a Tennessee Williams scholar at the Sewanee Writers' Conference and a James Merrill fellow at the Vermont Studio Center. She is an assistant professor of English and Creative Writing at Eastern New Mexico University.
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Please submit your news items to the Editor, Dr. Nancee Reeves (nreeves@uga.edu).
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redpajama_set_name: RedPajamaCommonCrawl


pelmetman - 2019-03-11 6:42 PM...……………….
1 Nope, I'm repeating what the Electoral Commission (EC) has said. Based on that, what I have said is that there is suspicion of electoral fraud which, if proved, would be illegal.
They are currently investigating the funding of the leave campaign, as there is evidence that at least some of its sources were illegal. So, declaring anything illegal at present would be premature. You really don't get this legal stuff, do you?
There has first to be reasonable grounds for suspicion. Tick. Then the grounds have to be investigated. Tick Based on the outcome, there has to be a review of the evidence to see if there is a case to be brought. No tick yet. If there is a case, it has to be tried. No tick yet. If it is tried, and is proved, it can then be said that the acts were illegal. No tick yet. All the EC is saying at the moment is that they smell a rat.
2 If there is evidence that the government used excessive funding, or used foreign funds, I would expect the EC to investigate that in exactly the same way as they are investigating the leave campaign funding. But, it would make no difference to the end result because, whichever side was found to have acted fraudulently, the referendum result would be suspect and, with any luck, void.
No Dave I say that there was nothing unlawful about the 9m spent on the government leaflet because it is fact.
You're clutching out for a lifebelt with pathetic pedantry instead of comprehending Brians very clear explanation.
Drop the silly pedantry and read Brians points 1 and 2 plus bear in mind Banks is still under criminal investigation by the NCA.
1 But as I pointed out, I have not made that claim, so how could I give evidence for something I haven't said? Can you think of some other things that I havent said, that I would be unable to prove? Is there a point?
2 How you choose to portray the EC and the government, or whatever names you call them, will not alter the outcomes of the investigations, will it?
I note you haven't been looking at the EC website to find the evidence you are asking others to provide for you. Why not? It isn't difficult, and it's very revealing! Just like the rest of us, you'll have to wait and see what happens. Its getting interesting, though!
But, as I keep telling you, it won't say that because the investigations are not yet complete. This is the UK, not the wild west!
We tend to gather evidence, evaluate it, bring charges, and try people - before finding them guilty. Can't imagine why, but think it may have something to with the rule of law. But then, you aren't in favour of that, are you? Odd!
I think you are confusing opinion with legal process.
Processing logic with a bloke who voted Brexit because he wants to "think he's British" is a futile task.
Can we all agree that it's time to rename Remain & Leave?
I suggest Right & Wrong.
I found the link to the full written judgement of Court of Appeal in the Wilson case dated 4.3.2019 for anyone who wants to read it and hasn't found it themselves.
In short it was not established that the referendum was illegal or that it had been shown that the illegalites identified by the Law Commision were material in relation to the outcome. Just as importantly perhaps the Court saw the challenge through the courts as "politics by other means" and that it was inappropriate for the Courts to interfere with the political process. Parliament had taken control of Brexit and it is parliament that has the inalienable right to make its own decision about whether the Law Commision's findings should be a consideration in its decisions on how and if Brexit should go ahead.
Can't see that leave to appeal to the Supreme Court was requested by those acting for Wilson et al.
Enemies of the people eh? I think not.
Sorry FP does this refer to my post or another one? I can't work it out.
I get it now that you have posted the terms of the Referendum Bill. No reply needed FP. Thanks.
No I have a problem with liars like yourself, who say it was a legally binding referendum when it wasn't.
No, but some of us find calling the opinion of 37% of the people "the will of the people" deliciously Stalinist! Joe would be really proud of you Comrade!
A turn out of 72% - higher than in many General Election and most certainly higher than any European Election.
I wonder how many of those who voted Remain were Irish Citizens?
How many who voted Remain ever voted in a European Parliament Election? Given that the turnout in the 2014 election was only 32.6%, not many.
I have to admit to being responsible for one of the spoiled votes in 2016. I honestly could not decide so decided to spoil my vote and go along with the majority decision.
Whether the Referendum was advisory or not and whether it was illegal is now irrelevant. Parliament voted to instigate Article 50, so like it or not, it is now Law.
That still won't alter the 37% Leave figure i'm afraid Pam. Brexiters incessant parroting of "will of the people" would be more honest if they began saying they want to see "the will of 37% of the electorate honoured"......but that doesn't quite have the same 'appeal' does it?
That percent figure will also now have undoubtedly decreased due to the time gap which has brought more first time voters in who couldn't vote at the time of the referendum, not to mention the vast devastation we've seen post referendum with businesses relocating and the countries economy trashed. This was why a peoples vote was always feared by Brexiters as they saw their "win" (because that's all what was important to them....'we won, you losers lost, get over it' etc) was in serious peril.
I'm surprised to see you feel the legality which is still under scrutiny, as 'irrelevant'. If guilt of meddling and wrong doings is found proven i could never be comfortable in knowing democracy had been subverted by external 'interference' to achieve a political aim.
Not much point in asking a question, and then giving the answer, is there/? Especially when you get the answer wrong.
I'm not interested in "winning", and have no desire to gloat and parade my tiny "victory" as a way to annoy those who voted to leave. There are more important things in life.
If Brexit is reversed, it will be the whole United Kingdom (including Brexiters) that "wins", and probably our shattered political system as well.
Hypocrisy? What "hypocrisy"? Lambaste the Brexiteer's "victory"? How? Where is the "shame" in wanting the whole UK to benefit from an economically sensible decision? What is good about inflicting economic loss on an entire country? In pursuit of what? Imaginary fears?


*Try and grab something sturdy and then bring a light DB from your side to 90 degrees for 10-12 reps on each arm. (1 arm at a time). Probably about 15-25lb Db range for guys and 5-15lb for ladies. Make it burn.
8 Box Jumps 24/20"4 DB Clean and Jerks on each side then switch. 70/50 is heavy, but if you get a good swing from the hip and a good knee drive on the push up, you should crush these. Legs will be sore from yesterday.
8 Lateral Burpees Over the bar or KB3min. Rest Between Each Section.
Section 2: If you can get 25 swings or swing a heavier KB, go for it!


موقع هذا المكان ممتاز وهو من الناحية الاقتصادية. لا شكاوى الرئيسية. الإفطار كان ممتازاً! مياه حمام السباحة كان حوض استحمام ساخن في تموز/يوليه. إنه فندق أقدم لذا لا تتوقع أن والرفاهية الحديثة.
Are there resort fees or a charge to park a car?
No resort fee for Las Brisas or parking fees.
Is there an elevator to get to the rooms off the first floor?
Yes. All floors can be reached by elevator.
Hello, We do allow bottled water poolside. No outside food and beverage as we serve lunch and appetizers poolside until 8PM daily.
Do the rooms have balconies or patios?