Abstract
§ 1. IntrodiictioE In the previous paper [8] we established a definition of an infinite tensor product ®*t of operators on ®J^t and studied its properties under the assumption: Ilil^JI <+ °°- In some applications, for instance, to Tomita's theory [3] and to quantum field theory [13, 14], we are obliged to work with a weaker assumption. In the present paper, we shall define an infinite tensor product ®C'CXL of operators xt on jf? as a closed linear mapping from an incomplete infinite tensor product space ®cJft to another ®c'jf4. We do not make any assumption on ||xj, allowing unbounded closed operators XL. The crucial assumption on (xt) is the existence of what we call a non-zero reference pair. This assumption turns out to be sufficiently general to allow various applications and yet sufficiently strong to yield significant results. Typical result is the following:
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