本文關(guān)鍵詞： 標(biāo)準(zhǔn)算子代數(shù) 不定內(nèi)積空間 因子von Neumann代數(shù) 斜Jordan零積 斜交換性 完全保持問題　出處：《太原科技大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：算子之間的斜Jordan零積、斜交換性等性質(zhì)特征在數(shù)學(xué)領(lǐng)域、量子力學(xué)和密碼學(xué)等領(lǐng)域中都有著廣泛的實際應(yīng)用背景.因此,越來越多的學(xué)者在保持問題框架下對算子之間的斜交換性等性質(zhì)進(jìn)行了研究.對保持問題的研究主要是對算子代數(shù)或算子空間上保持某種不變量(某種性質(zhì)、子集或關(guān)系等)的映射進(jìn)行研究,從而刻畫出該映射的具體結(jié)構(gòu)形式.在不同的算子空間或算子代數(shù)上對保持問題進(jìn)行討論,成為了泛函分析和算子代數(shù)理論上非�；钴S的研究課題,且獲得了一系列深刻的成果.近年來,隨著研究的不斷深入,完全保持問題的思想被眾多的學(xué)者深入探討.在已有的研究成果的基礎(chǔ)上,本文以算子的斜Jordan零積和斜交換性作為不變量,分別在標(biāo)準(zhǔn)算子代數(shù)、不定內(nèi)積空間的標(biāo)準(zhǔn)算子代數(shù)和因子von Neumann代數(shù)上進(jìn)行研究.從而對保持這些不變量的一般映射進(jìn)行刻畫.本文的主要研究結(jié)果如下:1.討論了無限維復(fù)的Hilbert空間上的*-標(biāo)準(zhǔn)算子代數(shù)之間雙邊完全保斜Jordan零積和斜交換性的一般映射,并證明了此類映射是同構(gòu)或者是共軛同構(gòu)的常數(shù)倍;2.研究了不定內(nèi)積空間上的?-標(biāo)準(zhǔn)算子代數(shù)之間雙邊完全保不定斜Jordan零積和不定斜交換性的一般映射,結(jié)果表明此類映射是同構(gòu)或者是共軛同構(gòu)的常數(shù)倍;3.刻畫了無限維復(fù)的Hilbert空間上的因子von Neumann代數(shù)之間雙邊完全保斜Jordan零積和斜交換性的一般映射,并且給出了映射的具體結(jié)構(gòu).
[Abstract]:The properties of oblique Jordan zero product and skew commutativity between operators have a wide range of practical applications in the fields of mathematics quantum mechanics and cryptography. More and more scholars have studied the properties of oblique commutativity between operators under the framework of maintenance problems. The study of conservation problems is mainly to preserve some invariants (some properties) on operator algebra or operator space. In this paper, we study the mapping of subsets or relations, so as to characterize the specific structural form of the mapping, and discuss the preserving problem in different operator spaces or operator algebras. It has become a very active research topic in functional analysis and operator algebra theory, and has obtained a series of profound results. On the basis of existing research results, this paper takes the oblique Jordan zero product and skew commutativity of operators as invariants, respectively, in the standard operator algebra. In this paper, we study the algebras of standard operators and factor von Neumann algebras of indefinite inner product spaces and characterize the general mappings that preserve these invariants. The main results of this paper are as follows:. 1. In this paper, we discuss the general mappings of two-sided completely skew preserving Jordan zero product and skew commutativity between two-sided completely skew Jordan algebras on infinite dimensional complex Hilbert spaces. It is proved that this kind of mapping is constant times of isomorphism or conjugate isomorphism. 2. In this paper, we study the indeterminate inner product space? General mappings between standard operator algebras which are completely indeterminate Jordan zero product and indeterminate commutativity. The results show that such mappings are constant times of isomorphism or conjugate isomorphism. 3. In this paper, we characterize the general mappings of two-sided completely oblique Jordan zero product and oblique commutativity between factor von Neumann algebras on infinite dimensional complex Hilbert spaces. The specific structure of the mapping is also given.
【學(xué)位授予單位】：太原科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O177

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