Operator Algebras, Interpolation and Frames

【摘要】保尔森：我的研究将集中在三个方向。我将继续研究注入包络，寻求关于其结构的更深入的结果，并进一步将这些对象应用于算子空间和C*-代数的研究。我将特别着重于尝试对注入包络和弱期望性质之间的联系进行更深入的理解。我还将继续研究算子代数技术在Nevanlinna-Pick型插值研究中的应用，特别是关注C*包络和舒尔理想的使用。我将重点研究的第三个方向是从编码理论的角度研究有限维框架，特别是确定某些编码理论问题的最优框架问题。-我对框架的研究可能会产生广泛的影响。帧被用作在存在不确定性的情况下确保模拟信息准确通信的一种方式，通常来自有噪声的通信信道。从编码理论的角度来看，我对帧的研究旨在找到用于传输数据的最佳帧，当预计一些数据将在传输过程中丢失时。我的工作与“包裹丢失”的问题特别相关。这里假设数据被打包成离散的“束”，然后分别传输，甚至可能通过不同的通道传输，然后再重新组装。问题是如果只有一部分包裹到达怎么办？显然，人们希望在传输的数据中建立一些冗余，以便仍然可以重新组装消息，即使不完全是原始消息，也可以在某种意义上接近原始消息。

AbstractPaulsenMy research will focus in three directions. I will continue to study injective envelopes, seeking deeper results about their structure and further applications of these objects to the study of operator spaces and C*-algebras. I will especially focus on attempting to develop a deeper understanding of the connections between injective envelopes and the weak expectation property. I will also continue to study applications of operator algebra techniques to the study of Nevanlinna-Pick type interpolation, especially focusing on the use of the C*-envelope and Schur ideals. The third direction that I will focus is the study of finite dimensional frames from the viewpoint of coding theory, especially the problem of determining optimal frames for certain coding theory problems.-My research on frames could have the broadest impact. Frames are used as a way to ensure accurate communication of analogue information in the presence of uncertainty, usually coming from a noisy communication channel. My work on frames from a coding theory point of view intends to find the optimal frames to use for transmitting data, when it is anticipated that some of the data will be lost during the transmission. My work is especially relevant to the problem of "lost packages". Here one assumes that the data is packaged into discrete "bundles" that are then transmitted separately, perhaps even over different channels, and then reassembled later. The problem is what to do if only some of the bundles arrive? Clearly, one wants to build some redundancy into the transmitted data so that one can still reassemble a message, that if not exactly the original message, is close to the original message in some sense.